Diffusion equations from master equations -- A discrete geometric   approach --

Shin-itiro Goto; Hideitsu Hino

arXiv:1908.04535·math-ph·November 6, 2020

Diffusion equations from master equations -- A discrete geometric approach --

Shin-itiro Goto, Hideitsu Hino

PDF

TL;DR

This paper reformulates continuous-time master equations in nonequilibrium statistical mechanics using discrete geometric methods, revealing their equivalence to diffusion equations and analyzing their spectral and convergence properties.

Contribution

It introduces a discrete geometric framework for master equations, establishing their connection to diffusion equations and providing new tools for analyzing their dynamics.

Findings

01

Master equations under detailed balance are equivalent to discrete diffusion equations.

02

The Laplacians in these diffusion equations are self-adjoint and have an isospectral property.

03

Convergence to equilibrium can be analyzed through the spectral properties of these Laplacians.

Abstract

In this paper, continuous-time master equations with finite states employed in nonequilibrium statistical mechanics are formulated in the language of discrete geometry. In this formulation, chains in algebraic topology are used, and master equations are described on graphs that consist of vertexes representing states and of directed edges representing transition matrices. It is then shown that master equations under the detailed balance conditions are equivalent to discrete diffusion equations, where the Laplacians are defined as self-adjoint operators with respect to introduced inner products. An isospectral property of these Laplacians is shown for non-zero eigenvalues, and its applications are given. The convergence to the equilibrium state is shown by analyzing this class of diffusion equations. In addition, a systematic way to derive closed dynamical systems for expectation values…

Equations490

\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&&\\&&&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 33.21704pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 72.52661pt\raise-4.50694pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.50694pt\hbox{$\scriptstyle{e}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 110.09557pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 74.43408pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 110.09557pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet}$}}}}}}}{\hbox{\kern-3.0pt\raise-15.22221pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 27.0pt\raise-15.22221pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{{\mathrm{o}}({e})}$}}}}}}}{\hbox{\kern 74.43408pt\raise-15.22221pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 104.43408pt\raise-15.22221pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{{\mathrm{t}}({e})}$}}}}}}}\ignorespaces}}}}\ignorespaces,\qquad\mbox{and}\quad\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&&\\&&&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 32.83336pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet}$}}}}}}}{\hbox{\kern 73.66672pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 110.05563pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 71.4445pt\raise-5.83888pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.83888pt\hbox{$\scriptstyle{\overline{e}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 43.83337pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern-3.0pt\raise-15.22221pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 27.0pt\raise-15.22221pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{{\mathrm{t}}({\overline{e}})}$}}}}}}}{\hbox{\kern 73.66672pt\raise-15.22221pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 103.66672pt\raise-15.22221pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{{\mathrm{o}}({\overline{e}})}$}}}}}}}\ignorespaces}}}}\ignorespaces,

\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&&\\&&&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 33.21704pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 72.52661pt\raise-4.50694pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.50694pt\hbox{$\scriptstyle{e}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 110.09557pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 74.43408pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 110.09557pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet}$}}}}}}}{\hbox{\kern-3.0pt\raise-15.22221pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 27.0pt\raise-15.22221pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{{\mathrm{o}}({e})}$}}}}}}}{\hbox{\kern 74.43408pt\raise-15.22221pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 104.43408pt\raise-15.22221pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{{\mathrm{t}}({e})}$}}}}}}}\ignorespaces}}}}\ignorespaces,\qquad\mbox{and}\quad\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&&\\&&&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 32.83336pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet}$}}}}}}}{\hbox{\kern 73.66672pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 110.05563pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 71.4445pt\raise-5.83888pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.83888pt\hbox{$\scriptstyle{\overline{e}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 43.83337pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern-3.0pt\raise-15.22221pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 27.0pt\raise-15.22221pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{{\mathrm{t}}({\overline{e}})}$}}}}}}}{\hbox{\kern 73.66672pt\raise-15.22221pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 103.66672pt\raise-15.22221pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{{\mathrm{o}}({\overline{e}})}$}}}}}}}\ignorespaces}}}}\ignorespaces,

t (\overline{e}) = o (e), \mbox an d o (\overline{e}) = t (e) .

t (\overline{e}) = o (e), \mbox an d o (\overline{e}) = t (e) .

C_{0} (G, R)

C_{0} (G, R)

C_{1} (G, R)

e \in E \sum a_{e} e = \overline{e} \in E \sum a_{\overline{e}} \overline{e} .

e \in E \sum a_{e} e = \overline{e} \in E \sum a_{\overline{e}} \overline{e} .

\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&&&&\\&&&&&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 28.79428pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern 62.37024pt\raise-11.83888pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.83888pt\hbox{$\scriptstyle{\overline{e^{\,\prime}}}$}}}\kern 3.0pt}}}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern 95.9541pt\raise-1.80237pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}{\hbox{\kern 65.58856pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 95.9462pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern 62.02391pt\raise 11.0625pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.0625pt\hbox{$\scriptstyle{e^{\,\prime}}$}}}\kern 3.0pt}}}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern 39.7865pt\raise 1.80237pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern 129.01083pt\raise 11.0625pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.0625pt\hbox{$\scriptstyle{e^{\,\prime\prime}}$}}}\kern 3.0pt}}}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern 163.87103pt\raise 1.7852pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}{\hbox{\kern 131.30383pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 163.8681pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern 129.90717pt\raise-11.83888pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.83888pt\hbox{$\scriptstyle{\overline{e^{\,\prime\prime}}}$}}}\kern 3.0pt}}}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern 106.94336pt\raise-1.7852pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}{\hbox{\kern-3.0pt\raise-13.15276pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 27.0pt\raise-13.15276pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{x^{\,\prime}}$}}}}}}}{\hbox{\kern 65.58856pt\raise-13.15276pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 95.58856pt\raise-13.15276pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{x}$}}}}}}}{\hbox{\kern 131.30383pt\raise-13.15276pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 161.30383pt\raise-13.15276pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{x^{\,\prime\prime}}$}}}}}}}\ignorespaces}}}}\ignorespaces.

\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&&&&\\&&&&&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 28.79428pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern 62.37024pt\raise-11.83888pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.83888pt\hbox{$\scriptstyle{\overline{e^{\,\prime}}}$}}}\kern 3.0pt}}}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern 95.9541pt\raise-1.80237pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}{\hbox{\kern 65.58856pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 95.9462pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern 62.02391pt\raise 11.0625pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.0625pt\hbox{$\scriptstyle{e^{\,\prime}}$}}}\kern 3.0pt}}}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern 39.7865pt\raise 1.80237pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern 129.01083pt\raise 11.0625pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.0625pt\hbox{$\scriptstyle{e^{\,\prime\prime}}$}}}\kern 3.0pt}}}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern 163.87103pt\raise 1.7852pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}{\hbox{\kern 131.30383pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 163.8681pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern 129.90717pt\raise-11.83888pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.83888pt\hbox{$\scriptstyle{\overline{e^{\,\prime\prime}}}$}}}\kern 3.0pt}}}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern 106.94336pt\raise-1.7852pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}{\hbox{\kern-3.0pt\raise-13.15276pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 27.0pt\raise-13.15276pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{x^{\,\prime}}$}}}}}}}{\hbox{\kern 65.58856pt\raise-13.15276pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 95.58856pt\raise-13.15276pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{x}$}}}}}}}{\hbox{\kern 131.30383pt\raise-13.15276pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 161.30383pt\raise-13.15276pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{x^{\,\prime\prime}}$}}}}}}}\ignorespaces}}}}\ignorespaces.

y \in V \sum a_{y} y = a_{x^{'}} x^{'} + a_{x} x + a_{x^{''}} x^{''} .

y \in V \sum a_{y} y = a_{x^{'}} x^{'} + a_{x} x + a_{x^{''}} x^{''} .

e \in E \sum a_{e} e = a_{e^{'}} e^{'} + a_{e^{''}} e^{''} + a_{\overline{e^{'}}} \overline{e^{'}} + a_{\overline{e^{''}}} \overline{e^{''}} = \overline{e} \in E \sum a_{\overline{e}} \overline{e} .

e \in E \sum a_{e} e = a_{e^{'}} e^{'} + a_{e^{''}} e^{''} + a_{\overline{e^{'}}} \overline{e^{'}} + a_{\overline{e^{''}}} \overline{e^{''}} = \overline{e} \in E \sum a_{\overline{e}} \overline{e} .

C^{0} (G, R)

C^{0} (G, R)

C^{1} (G, R)

Λ^{1} (G, R)

Λ^{1} (G, R)

S^{1} (G, R)

\partial : C_{1} (G) \to C_{0} (G), \mbox so t ha t \partial (e) := t (e) - o (e),

\partial : C_{1} (G) \to C_{0} (G), \mbox so t ha t \partial (e) := t (e) - o (e),

\partial (c_{1} + c_{2}) = \partial c_{1} + \partial c_{2}, \mbox an d \partial (a c) = a \partial c, a \in R, c, c_{1}, c_{2} \in C_{1} (G) .

\partial (c_{1} + c_{2}) = \partial c_{1} + \partial c_{2}, \mbox an d \partial (a c) = a \partial c, a \in R, c, c_{1}, c_{2} \in C_{1} (G) .

d : Λ^{0} (G) \to Λ^{1} (G), \mbox so t ha t (d f) (e) := f (t (e)) - f (o (e)),

d : Λ^{0} (G) \to Λ^{1} (G), \mbox so t ha t (d f) (e) := f (t (e)) - f (o (e)),

d (f_{1} + f_{2}) = d f_{1} + d f_{2}, \mbox an d d (a f) = a d f, a \in R, f, f_{1}, f_{2} \in Λ^{0} (G) .

d (f_{1} + f_{2}) = d f_{1} + d f_{2}, \mbox an d d (a f) = a d f, a \in R, f, f_{1}, f_{2} \in Λ^{0} (G) .

(d f) (\overline{e}) = f (t (\overline{e})) - f (o (\overline{e})) = f (o (e)) - f (t (e)) = - (d f) (e) .

(d f) (\overline{e}) = f (t (\overline{e})) - f (o (\overline{e})) = f (o (e)) - f (t (e)) = - (d f) (e) .

⟨, ⟩_{V} : Λ^{0} (G) \times Λ^{0} (G) \to R \mbox an d ⟨, ⟩_{E} : Λ^{1} (G) \times Λ^{1} (G) \to R,

⟨, ⟩_{V} : Λ^{0} (G) \times Λ^{0} (G) \to R \mbox an d ⟨, ⟩_{E} : Λ^{1} (G) \times Λ^{1} (G) \to R,

m_{V} (x) > 0, \forall x \in V,

m_{V} (x) > 0, \forall x \in V,

m_{E} (e) = m_{E} (\overline{e}) > 0, \forall e \in E .

m_{E} (e) = m_{E} (\overline{e}) > 0, \forall e \in E .

⟨ f_{1}, f_{2} ⟩_{V}

⟨ f_{1}, f_{2} ⟩_{V}

⟨ ω_{1}, ω_{2} ⟩_{E}

(d^{†} ω) (x) = \frac{- 1}{m _{V} ( x )} e \in E_{x} \sum ω (e) m_{E} (e),

(d^{†} ω) (x) = \frac{- 1}{m _{V} ( x )} e \in E_{x} \sum ω (e) m_{E} (e),

E_{x} := {e \in E ∣ o (e) = x} .

E_{x} := {e \in E ∣ o (e) = x} .

d^{†} (ω_{1} + ω_{2}) = d^{†} ω_{1} + d^{†} ω_{2}, \mbox an d d^{†} (a ω) = a d^{†} ω, a \in R, ω, ω_{1}, ω_{2} \in Λ^{1} (G) .

d^{†} (ω_{1} + ω_{2}) = d^{†} ω_{1} + d^{†} ω_{2}, \mbox an d d^{†} (a ω) = a d^{†} ω, a \in R, ω, ω_{1}, ω_{2} \in Λ^{1} (G) .

e \in E \sum \dots = x \in V \sum e \in E_{x} \sum \dots

e \in E \sum \dots = x \in V \sum e \in E_{x} \sum \dots

\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&&&&\\&&&&&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 28.79428pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern 62.37024pt\raise-11.83888pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.83888pt\hbox{$\scriptstyle{\overline{e^{\,\prime}}}$}}}\kern 3.0pt}}}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern 95.9541pt\raise-1.80237pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}{\hbox{\kern 65.58856pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 95.9462pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern 62.02391pt\raise 11.0625pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.0625pt\hbox{$\scriptstyle{e^{\,\prime}}$}}}\kern 3.0pt}}}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern 39.7865pt\raise 1.80237pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern 129.01083pt\raise 11.0625pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.0625pt\hbox{$\scriptstyle{e^{\,\prime\prime}}$}}}\kern 3.0pt}}}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern 163.87103pt\raise 1.7852pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}{\hbox{\kern 131.30383pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 163.8681pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern 129.90717pt\raise-11.83888pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.83888pt\hbox{$\scriptstyle{\overline{e^{\,\prime\prime}}}$}}}\kern 3.0pt}}}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern 106.94336pt\raise-1.7852pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}{\hbox{\kern-3.0pt\raise-13.15276pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 27.0pt\raise-13.15276pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{x^{\,\prime}}$}}}}}}}{\hbox{\kern 65.58856pt\raise-13.15276pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 95.58856pt\raise-13.15276pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{x}$}}}}}}}{\hbox{\kern 131.30383pt\raise-13.15276pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 161.30383pt\raise-13.15276pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{x^{\,\prime\prime}}$}}}}}}}\ignorespaces}}}}\ignorespaces.

\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&&&&\\&&&&&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 28.79428pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern 62.37024pt\raise-11.83888pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.83888pt\hbox{$\scriptstyle{\overline{e^{\,\prime}}}$}}}\kern 3.0pt}}}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern 95.9541pt\raise-1.80237pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}{\hbox{\kern 65.58856pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 95.9462pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern 62.02391pt\raise 11.0625pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.0625pt\hbox{$\scriptstyle{e^{\,\prime}}$}}}\kern 3.0pt}}}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern 39.7865pt\raise 1.80237pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern 129.01083pt\raise 11.0625pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.0625pt\hbox{$\scriptstyle{e^{\,\prime\prime}}$}}}\kern 3.0pt}}}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern 163.87103pt\raise 1.7852pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}{\hbox{\kern 131.30383pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 163.8681pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern 129.90717pt\raise-11.83888pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.83888pt\hbox{$\scriptstyle{\overline{e^{\,\prime\prime}}}$}}}\kern 3.0pt}}}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern 106.94336pt\raise-1.7852pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}{\hbox{\kern-3.0pt\raise-13.15276pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 27.0pt\raise-13.15276pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{x^{\,\prime}}$}}}}}}}{\hbox{\kern 65.58856pt\raise-13.15276pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 95.58856pt\raise-13.15276pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{x}$}}}}}}}{\hbox{\kern 131.30383pt\raise-13.15276pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 161.30383pt\raise-13.15276pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{x^{\,\prime\prime}}$}}}}}}}\ignorespaces}}}}\ignorespaces.

E_{x^{'}} = {\overline{e^{'}}}, E_{x} = {e^{'}, e^{''}}, E_{x^{''}} = {\overline{e^{''}}},

E_{x^{'}} = {\overline{e^{'}}}, E_{x} = {e^{'}, e^{''}}, E_{x^{''}} = {\overline{e^{''}}},

e \in E \sum ω (e)

e \in E \sum ω (e)

⟨ d f, ω ⟩_{E} = ⟨ f, d^{†} ω ⟩_{V} .

⟨ d f, ω ⟩_{E} = ⟨ f, d^{†} ω ⟩_{V} .

⟨ d f, ω ⟩_{E}

⟨ d f, ω ⟩_{E}

div := - d^{†} .

div := - d^{†} .

Peer Reviews

No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.

Videos

No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.

Full text

Diffusion equations from master equations

— A discrete geometric approach —

Shin-itiro GOTO and Hideitsu HINO

The Institute of Statistical Mathematics,

10-3 Midori-cho, Tachikawa, Tokyo 190-8562, Japan

Abstract

In this paper, continuous-time master equations with finite states employed in nonequilibrium statistical mechanics are formulated in the language of discrete geometry. In this formulation, chains in algebraic topology are used, and master equations are described on graphs that consist of vertexes representing states and of directed edges representing transition matrices. It is then shown that master equations under the detailed balance conditions are equivalent to discrete diffusion equations, where the Laplacians are defined as self-adjoint operators with respect to introduced inner products. An isospectral property of these Laplacians is shown for non-zero eigenvalues, and its applications are given. The convergence to the equilibrium state is shown by analyzing this class of diffusion equations. In addition, a systematic way to derive closed dynamical systems for expectation values is given. For the case that the detailed balance conditions are not imposed, master equations are expressed as a form of a continuity equation.

1 Introduction

Master equations are vital in the study of nonequilibrium statistical mechanics [1, 2, 3], since they are mathematically simple and show relaxation processes towards to equilibrium states under some conditions [4]. These equations describe the time evolution of probabilities, and they are first order differential or difference equations. In addition, these equations are used in Monte Carlo simulations [5]. Quantum mechanical case can be considered by extending classical systems [6]. Thus there have been a variety of applications in mathematical sciences, and its progress continues to attract attention in the literature [7, 8, 9, 10].

Algebraic topology and graph theory have been applied to a variety of sciences and mathematical engineering. Several topological approaches to master equations exist in the literature [11, 12, 13, 14, 15]. Not only master equations, but also random walks on lattices [16], electric circuits [17, 18, 19], and so on, have been studied from the viewpoint of algebraic topology. By introducing inner products for functions on graphs together with coboundary operator, one can define the adjoint of the coboundary operator, and Laplacians as self-adjoint operators [16]. Corresponding operators in the continuous case are useful as proven in the literature of functional analysis [20]. An amalgamation of these mathematical disciplines may be called discrete geometry. It is then of interest to explore how above-mentioned operators in discrete geometry can be used for master equations. Moreover, although discrete diffusion equations have been derived from master equations in some cases, the condition when such diffusion equations can be derived is not known. Since the knowledge of discrete diffusion equations has been accumulated, clarifying such a condition is expected to be fruitful for the study of master equations.

In this paper continuous-time master equations are formulated in terms of functions of chains, where states and transition matrices for master equations are described by chains used in algebraic topology. In particular, probability distribution function is regarded as a function of [math]-chain, and transition matrix a function of $1$ -chain, where a discrete state is expressed as a vertex or a [math]-chain. After introducing some inner products for functions on chains and current as a function on $1$ -chain, master equations are shown to be written in terms of the adjoint of the coboundary operator acting on the current:

*Claim**.*

Master equations can be written as a form of a continuity equation (See Theorem 3.1 for details).

In addition, under the assumption that the detailed balance conditions hold, an equivalence between master equations and discrete diffusion equations is shown, where the Laplacians are constructed by choosing appropriate measures for inner products:

*Claim**.*

Master equations under the detailed balance conditions are equivalent to discrete diffusion equations (See Theorem 3.2 for details).

By applying this statement, it is shown that probability distribution functions relax to the equilibrium state (See Corollary 3.1 and Proposition 3.2). Contrary to this, it is shown that discrete diffusion equations yield master equations (See Proposition 3.1). Meanwhile, an isospectral property for non-zero eigenvalues of these Laplacians is given (See Theorem 3.3), and this can be referred to as a supersymmetry [21]. With this supersymmetry dynamical systems for expectation variables are derived without any approximation (See Propositions 3.5— 3.8). Some inequalities are also derived (See Propositions 3.3 and 3.4).

These theorems, corollary, and so on should be compared with those in the existing literature. In the study of random walks on lattices, chains, functions and their Laplacians are also used [16]. In the literature the probability distribution functions are identified with functions on $1$ -chains, which is different to the present formalism. The differences appear since transition matrices are introduced for describing state transitions for the present study only. In contrast, similarities between the present study and existing studies appear due to the use of Laplacians. Laplacian is a self-adjoint operator with respect to an inner product, and brings several properties as well as the case of the standard Riemannian geometry.

The rest of this paper is organized as follows. In Section 2, some preliminaries are provided in order to keep this paper self-contained. They include boundary operators, inner products, and Laplacians. Most of such tools and notions have been described in Ref. [16], while others are invented in this paper. In Section 3, master equations are formulated as dynamical systems on graph, and the main claims of this paper and their consequences are provided. Namely, diffusion equations are derived by means of tools introduced in the previous section. Moreover, various inequalities and dynamical systems for expectation variables are derived. Section 4 summarizes this paper and discusses some future studies.

2 Preliminaries

Let $G=(V,E)$ be a directed graph with $V$ a vertex set and $E$ an edge set. Throughout this paper, every graph is finite ( $\#E<\infty$ ), connected, and allowed to have loop edges. In addition, an inverse edge $\overline{e}\in E$ is assumed to exist for each edge $e\in E$ . However parallel edges that will be defined later are excluded from this contribution. Details of these assumptions are explained in Section 2.1.

2.1 Standard operators

In this subsection, most of notions and notations follow Ref. [16]. However, some variants are also introduced.

For a given edge $e\in E$ , the inverse of $e$ , the terminus of $e$ , and the origin of $e$ are denoted by $\overline{e}$ , ${\mathrm{t}}({e})\in V$ , and ${\mathrm{o}}({e})\in V$ , respectively, as follows:

[TABLE]

where $\overline{e}$ has not been depicted in the left graph, and $e$ has not been depicted in the right graph above. Then, it follows that

[TABLE]

In this paper $\overline{e}\in E$ is assumed to exist for any edge $e\in E$ . Thus, even $\overline{e}$ is not depicted for a given $e$ , the existence of its inverse edge $\overline{e}$ is assumed in this section. A loop edge $e\in E$ is such that ${\mathrm{o}}({e})={\mathrm{t}}({e})$ , and parallel edges $e_{\,1},e_{\,2}\in E$ are such that $e_{\,1}\neq e_{\,2}$ with ${\mathrm{o}}({e_{\,1}})={\mathrm{o}}({e_{\,2}})$ and ${\mathrm{t}}({e_{\,1}})={\mathrm{t}}({e_{\,2}})$ . Parallel edges are assumed not to exist in any graph in this paper.

Then one defines the groups of [math]-chains and $1$ -chains on a graph $G$ with coefficients $\mathbb{R}$

[TABLE]

respectively. Here $\sum_{e\in E}a_{\,e}e$ denotes the sum over all $e\in E$ in the sense that

[TABLE]

As an example, consider the graph $G=(V,E)$ with

[TABLE]

In this case, with some $a_{\,x^{\,\prime}},a_{\,x},a_{\,x^{\,\prime\prime}}\in\mathbb{R}$ , an element of $C_{\,0}(G)$ is expressed as

[TABLE]

Then with some $a_{\,e^{\,\prime}},a_{\,e^{\,\prime\prime}},a_{\,\overline{e^{\,\prime}}},a_{\,\overline{e^{\,\prime\prime}}}\in\mathbb{R}$ , an element of $C_{\,1}(G)$ is expressed as

[TABLE]

The spaces of functions on $C^{\,0}(G,\mathbb{R})$ and $C^{\,1}(G,\mathbb{R})$ are denoted by

[TABLE]

Elements of the subset of $C^{\,0}(G,\mathbb{R})$ being dual to $C_{\,0}(G,\mathbb{R})$ are referred to as [math]-cochains. Here the dual space of a linear space is, by definition, a linear space in general. Similarly, elements of the subset of $C^{\,1}(G,\mathbb{R})$ being dual to $C_{\,1}(G,\mathbb{R})$ are referred to as $1$ -cochains. The set $C^{\,0}(G,\mathbb{R})$ is also denoted by $\Lambda^{\,0}(G,\mathbb{R})$ in this paper. Note that $f(x)\in\mathbb{R}$ with $f\in C^{\,0}(G,\mathbb{R})$ needs not be linear in $x$ , and that $\omega(e)\in\mathbb{R}$ with $\omega\in C^{\,1}(G,\mathbb{R})$ needs not be linear in $e$ . The subsets $\Lambda^{\,1}(G,\mathbb{R})\subset C^{\,1}(G,\mathbb{R})$ and $S^{\,1}(G,\mathbb{R})\subset C^{\,1}(G,\mathbb{R})$ are defined by

[TABLE]

In what follows, $C_{\,0}(G,\mathbb{R})$ and $C_{\,1}(G,\mathbb{R})$ are often abbreviated as $C_{\,0}(G)$ and $C_{\,1}(G)$ , respectively. Similar abbreviations will be adopted.

The boundary operator is defined as

[TABLE]

and the linearity holds:

[TABLE]

The dual of the boundary operator, the coboundary operator, is defined by

[TABLE]

and the linearity holds:

[TABLE]

If $f$ is a [math]-cochain, then it follows that $(\mathrm{d}f)(e)=f(\partial e)$ . For any loop edge $e\in E$ , it follows from $f({\mathrm{t}}({e}))=f({\mathrm{o}}({e}))$ that $(\mathrm{d}f)(e)=0$ . This $\mathrm{d}f$ is indeed an element belonging to $\Lambda^{\,1}(G)$ , as shown below. For any $f\in\Lambda^{\,0}(G)$ and $e\in E$ , one has that

[TABLE]

To define inner products,

[TABLE]

one introduces some measures. Let $m_{\,V}$ and $m_{\,E}$ be elements of $C^{\,0}(G)$ and $S^{\,1}(G)$ , such that

[TABLE]

and

[TABLE]

This $m_{\,E}\in S^{\,1}(G)$ is referred to as a reversible measure.

The following are inner products

[TABLE]

Associated with this set of inner products, one defines the adjoint of the coboundary operator $\mathrm{d}^{\dagger}:\Lambda^{\,1}(G)\to\Lambda^{\,0}(G)$ . This $\mathrm{d}^{\dagger}$ is also referred to as co-derivative in this paper, which is explicitly written as

[TABLE]

where

[TABLE]

It follows from (3) that the linearity holds:

[TABLE]

Notice that the decomposition of the sum

[TABLE]

holds. To show examples of (4) and (5), consider the graph $G=(V,E)$ ,

[TABLE]

For all vertexes, (4) is expressed as

[TABLE]

and the sum (5) with $\omega\in\Lambda^{\,1}(G)$ is expressed as

[TABLE]

The operator $\mathrm{d}^{\,\dagger}$ is indeed the adjoint one of $\mathrm{d}$ as shown below.

Lemma 2.1.

[TABLE]

Proof.

Straightforward calculations with (5) yield

[TABLE]

∎

Remark 2.1.

An operator analogous to $-\,\mathrm{d}^{\dagger}$ in the continuous standard Riemannian geometry is referred to as divergence. Thus, $-\,\mathrm{d}^{\dagger}$ can be referred to as *divergence * on graph [22]. This operator $\Lambda^{\,1}(G)\to\Lambda^{\,0}(G)$ is denoted by

[TABLE]

The Laplacian acting on $\Lambda^{\,0}(G)$ , $\Delta_{\,V}:\Lambda^{\,0}(G)\to\Lambda^{\,0}(G)$ , is defined as

[TABLE]

The explicit form of its action is obtained as follows. By putting $\omega=\mathrm{d}f$ for $f\in\Lambda^{\,0}(G)$ , one has

[TABLE]

Since $\mathrm{d}$ and $\mathrm{d}^{\dagger}$ are linear operators, $\Delta_{\,V}$ is also a linear operator:

[TABLE]

Moreover, it follows that

[TABLE]

To see a link between (8) and a well-known form of discrete Laplacian, choose $m_{\,V}(x)=\delta$ and $m_{\,E}(e)=1$ for all $x\in V$ and $e\in E$ , where $\delta>0$ is a constant. Then consider the graph

[TABLE]

where $\overline{e^{\,\prime}}$ and $\overline{e^{\,\prime\prime}}$ have been omitted. For this graph, one has the well-known form:

[TABLE]

By taking the limit $\delta\to 0$ appropriately, one has the second derivative of $f$ . See Ref. [24] for another link between this Laplacian $\Delta_{\,V}$ and the so-called adjacency operator.

The operator $\Delta_{\,V}$ is self-adjoint as shown below.

Lemma 2.2.

[TABLE]

Proof.

It follows that

[TABLE]

∎

The Laplacian acting on $\Lambda^{\,1}(G)$ , $\Delta_{\,E}:\Lambda^{\,1}(G)\to\Lambda^{\,1}(G)$ , is defined as

[TABLE]

This operator is self-adjoint as shown below.

Lemma 2.3.

[TABLE]

Proof.

It can be proven by straightforward calculations. To this end, we put $f_{\,1}=\mathrm{d}^{\,\dagger}\omega_{1}\in\Lambda^{\,0}(G)$ and then it follows that

[TABLE]

∎

Most of the operators and their properties discussed so far are well-known [16]. By contrast, those discussed in the next subsection are not standard ones.

2.2 Operators for master equations

To discuss how to describe master equations in the language of discrete geometry, one first introduces

[TABLE]

Then, with a prescribed $\varphi\in C_{\,{\mathrm{R}}}^{\,1}(G)$ , one introduces

[TABLE]

The operator associated with $\varphi\in C_{\,{\mathrm{R}}}^{\,1}(G)$ is defined as

[TABLE]

and the linearity in $f$ holds:

[TABLE]

Condition in (10) is to guarantee the property $\mathrm{d}_{\,\varphi}\,f\in C_{\,\varphi}^{\,1}(G)$ for $f\in C^{\,0}(G)$ ,

[TABLE]

The above equality is verified as

[TABLE]

Remark 2.2.

In the case that $\varphi$ is such that $\varphi(e)=1$ for any $e\in E$ , one has that $\mathrm{d}_{\,\varphi}\,f=\mathrm{d}f$ for any $f\in C^{\,0}(G)$ . The operator $\mathrm{d}_{\,\varphi}$ is an analogue of the one introduced in Ref. [23].

As well as the case for $\left\langle\,\,,\,\,\right\rangle_{\,E}$ , one introduces an inner product on $C_{\,\varphi}^{\,1}(G)$

[TABLE]

associated with a reversible measure $m_{\,E}\in S^{\,1}(G)$ . The value of the inner product is the same as (2):

[TABLE]

The inner product (12) should be notationally distinguished from (2), since (2) has been defined as $\left\langle\,\,,\,\,\right\rangle_{\,E}:\Lambda^{\,1}(G)\times\Lambda^{\,1}(G)\to\mathbb{R}$ . The inner product $\left\langle\,\,,\,\,\right\rangle_{\,E}^{\varphi}$ coincides with $\left\langle\,\,,\,\,\right\rangle_{\,E}$ when $\varphi(e)=1$ for all $e\in E$ . With (12), one introduces the co-derivative on $C_{\,{\mathrm{R}}}^{\,1}(G)$ , $\mathrm{d}_{\,\varphi}^{\dagger}:C_{\,\varphi}^{\,1}(G)\to C^{\,0}(G)$ with $\varphi\in C_{\,{\mathrm{R}}}^{\,1}(G)$ such that

[TABLE]

Lemma 2.4.

For $\varphi\in C_{\,{\mathrm{R}}}^{\,1}(G),\ \omega\in C_{\,\varphi}^{\,1}(G),$ and $f\in C^{\,0}(G)$ , one has

[TABLE]

Proof.

Substituting

[TABLE]

into

[TABLE]

one has

[TABLE]

∎

Although the following operators will not be used in Section 3, the Laplacians are discussed for the sake of completeness. The Laplacian $\Delta_{\,V}^{\,\varphi}:C^{\,0}(G)\to C^{\,0}(G)$ is defined as

[TABLE]

The explicit form of its action is obtained as follows. By putting $\omega=\mathrm{d}_{\,\varphi}f$ for $f\in C^{\,0}(G)$ , one has

[TABLE]

This operator is self-adjoint:

[TABLE]

since

[TABLE]

In addition, the Laplacian $\Delta_{\,E}^{\,\varphi}:C_{\,\varphi}^{\,1}\to C_{\,\varphi}^{\,1}$ is defined as

[TABLE]

This operator is self-adjoint:

[TABLE]

since

[TABLE]

3 Master equations

Master equations are used to model nonequilibrium time-development in various systems [4, 11], and they are written as

[TABLE]

where $\{x\}$ are discrete states, $p_{\,t}(x)$ a probability distribution function of $x$ at time $t\in\mathbb{R}$ , $w_{\,x\to x^{\prime}}$ a transition matrix that describes the transition rate from a state $x$ to another state $x^{\,\prime}$ . In addition, the set $\{w_{x\to x^{\prime}}\}$ is chosen so that the conservation law $\sum_{x}p_{\,t}(x)=1$ holds, and is used to model physical process of a system under consideration. In this paper we focus on the case where $\{w_{x\to x^{\prime}}\}$ does not depend on time. The equilibrium state at $x$ is denoted by $p^{\,\mathrm{eq}}(x)$ . In what follows, these objects, $\{x\}$ , $\{w_{x\to x^{\prime}}\}$ , and $\{p^{\,\mathrm{eq}}(x)\}$ , are treated as given data. Although there are several closely related equations of (13), including backward master equations, discrete-time master equations, and so on [3], they are not addressed in this paper.

In this section, a graph formulation of master equations is shown in terms of objects developed in Section 2. Main claims of this paper and their consequences are then provided.

3.1 Graph formulation

Introduce a graph $G=(V,E)$ and $w\in C^{\,1}(G)$ associated with the given data $\{x\}$ and $\{w_{\,x\to x^{\prime}}\}$ such that the following hold :

•

For any $x$ , $x$ is an element of $V$ , that is, $x\in V$ .

•

Let $w\in C^{\,1}(G)$ be such that $w(e)=w_{\,x\to x^{\prime}}$ for $e\in E$ with ${\mathrm{t}}({e})=x^{\,\prime}$ and ${\mathrm{o}}({e})=x$ .

•

For any $e\in E$ , its inverse exists, that is, $\overline{e}\in E$ . If $w(\overline{e})=w_{\,y\to x}$ does not exist, then let $w$ be such that $w(\overline{e})=0$ .

Then, regard $p_{\,t}$ as follows :

•

$p_{\,t}\in\Lambda^{\,0}(G)$ so that $p_{\,t}(x)\in\mathbb{R}$ for any $x\in V$ .

The master equations (13) can be written in terms of these graph terminologies as

[TABLE]

where

[TABLE]

The correspondence between (14) and (13) is obvious. Vertexes ${\mathrm{o}}({e})$ and ${\mathrm{t}}({e})$ in (14) correspond to states $x$ and $x^{\,\prime}$ in (13), and the first $w(e)\in\mathbb{R}$ and second $w(e)\in\mathbb{R}$ in (14) correspond to $w_{\,x\to x^{\,\prime}}$ and $w_{\,x^{\,\prime}\to x}$ in (13), respectively. Notice that the conservation law for (13) translates into $\sum_{x\in V}p_{\,t}(x)=1$ for (14). By construction of the present graph formulation, (14) reduces further. Given an $x\in V$ and $e\in E^{\,x}$ , there exists $e^{\,\prime}\in E_{\,x}$ , such that

[TABLE]

It then follows that ${\mathrm{o}}({e})={\mathrm{t}}({e^{\,\prime}})$ and $e=\overline{e^{\,\prime}}$ , from which

[TABLE]

Then (14) reduces to

[TABLE]

An example of this set of the procedures is given below.

Example 3.1.

(A two-state model). Consider the set of master equations whose total number of states is $2$ ,

[TABLE]

where $w_{\,1\to 2}$ is a positive constant. This set of equations can be represented in terms of the graph

[TABLE]

where the edge $e^{\,\prime\prime}$ has been added such that $\overline{e^{\,\prime}}$ exists. A consistency between this graph expression and the given set of master equations is verified as follows. Since $\{e^{\,\prime}\}=E_{\,x_{1}}$ , $\{e^{\,\prime\prime}\}=E_{\,x_{2}}$ , $x_{\,1}={\mathrm{o}}({e^{\,\prime}})={\mathrm{t}}({e^{\,\prime\prime}})$ , $x_{\,2}={\mathrm{t}}({e^{\,\prime}})={\mathrm{o}}({e^{\,\prime\prime}})$ , and $\overline{e^{\,\prime\prime}}=e^{\,\prime}$ , (15) yields

[TABLE]

and

[TABLE]

For completeness, the solution satisfying the conservation law $p_{\,t}(x_{\,1})+p_{\,t}(x_{\,2})=1$ for all $t\in\mathbb{R}$ is explicitly shown as

[TABLE]

where $p_{\,0}(x_{\,1})$ and $p_{\,0}(x_{\,2})$ are constants satisfying $0\leq p_{\,0}(x_{\,1})\leq 1$ and $p_{\,0}(x_{\,2})=1-p_{\,0}(x_{\,1})$ .

In what follows, master equations are rewritten in the language discussed in Section 2. To this end, one defines $J_{\,t}\in C^{\,1}(G)$ and ${\cal I}_{\,t}\in\Lambda^{\,1}(G)$ so that

[TABLE]

One verifies that ${\cal I}_{\,t}(\overline{e})=-{\cal I}_{\,t}(e)$ immediately, and that $-{\cal I}_{\,t}(e)$ is the summand in the right hand side of (15),

[TABLE]

and thus,

[TABLE]

The physical meaning of the case of ${\cal I}_{\,t}(e)=0$ for a given $e\in E$ is that the probability flow or current is locally balanced. If the condition $w(\overline{e})=w(e)$ is satisfied for all $e\in E$ , then one has

[TABLE]

This relation corresponds to Fick’s law of diffusion in continuous media [1].

To describe some measures, introduce

[TABLE]

In terms of these objects, one has the following:

Theorem 3.1.

*(Continuity equation as master equations).

Choose $m_{\,V}\in\Lambda^{\,0}(G)$ and $m_{\,E}\in S^{\,1}(G)$ to be*

[TABLE]

Then (15) is identical to

[TABLE]

Proof.

With the choices $m_{\,V}$ and $m_{\,E}$ , the co-derivative $\mathrm{d}^{\dagger}$ defined in (3) reduces to the one such that

[TABLE]

From this and (15), one has

[TABLE]

for any $x\in V$ . Thus, the desired expression is obtained. ∎

It is worth mentioning that (18) is written as a form of a continuity equation in continuum mechanics

[TABLE]

where $\mathrm{div}:\Lambda^{\,1}(G)\to\Lambda^{\,0}(G)$ has been defined in (6). In continuum mechanics a continuity equation for a time-dependent scalar quantity with an associated current leads to a conservation law by integration over a spatial region. Correspondingly, the equivalent form (19) of (15) leads to a conserved law by summing $p_{\,t}$ over all states (See Remark 3.1). Besides, in the study of master equations the form similar to (19) is well-known in another formulation (See Refs. [14, 15] for example).

Theorem 3.1 deals with $p_{\,t}\in\Lambda^{\,0}(G)$ . By contrast, an equality on $\Lambda^{\,1}(G)$ associated with the master equations is obtained from (18) as

[TABLE]

where $\Delta_{\,E}$ has been defined in (9).

An example is shown below so that how introduced objects can be calculated from a given set of master equations, and that how (18) is applied.

Example 3.2.

(A one-way interaction model on a ring). Consider the set of master equations whose total number of states is $N<\infty$ ,

[TABLE]

where $x_{\,N+j}=x_{\,j}$ so that $p_{\,t}(x_{\,N+j})=p_{\,t}(x_{\,j})$ for all $t\in\mathbb{R}$ , and $w_{j\to j+1}$ is a positive constant for each $j$ . This model expresses a class of the so-called one-step processes [4], and can be represented as the graph $G^{\,\prime}=(V^{\,\prime},E^{\,\prime})$ depicted as follows:

[TABLE]

In the graph above, there is no $\overline{e}$ for a given $e\in E^{\,\prime}$ . Introducing the edge set $E$ from $E^{\,\prime}$ such that there exists $\overline{e}\in E$ for all $e\in E$ , one has the graph $G=(V,E)$ with $V=V^{\,\prime}$ depicted as

[TABLE]

To gain physical insight by analyzing a simple case, all the constants $\{w_{\,j-1\to j}\}$ are assumed to be equal, which is denoted by $\mu>0$ . Then, to simplify the notation, $p_{\,t}(x_{\,j})$ is abbreviated to $p_{\,t}(j)$ so that (21) is written as

[TABLE]

Introduce $w\in C^{\,1}(G)$ such that

[TABLE]

From (19), the divergence of ${\cal I}_{\,t}$ at $x_{\,j}$ is expressed as

[TABLE]

For completeness, the solution satisfying the conservation law $\sum_{j=0}^{N-1}p_{\,t}(x_{\,j})=1$ for all $t\in\mathbb{R}$ is shown. First, introduce the Fourier transform of $p_{\,t}(j)$ with respect to $j$ as

[TABLE]

From this expression of $\widehat{p}_{\,t}(k)$ , it is straightforward to show that

[TABLE]

Second, it follows that the Fourier transformed variables $\widehat{p}_{\,t}(k)$ satisfy

[TABLE]

The solution to its initial value problem is immediately obtained as

[TABLE]

which contains the conservation law at $k=0$ ,

[TABLE]

Finally, the solution to the initial value problem in terms of the original variables is

[TABLE]

3.1.1 Expectation values

Much attention is devoted to expectation values with respect to $p_{\,t}$ in applications of master equations. To discuss such expectation values, one introduces ${\cal O}^{\,0}\in\Lambda^{\,0}(G)$ that is to be summed over vertexes at fixed time. In this subsection, $m_{\,V}\in\Lambda^{\,0}(G)$ and $m_{\,E}\in S^{\,1}(G)$ are chosen as (17).

The expectation value of ${\cal O}^{\,0}$ with respect to $p_{\,t}$ is denoted by

[TABLE]

which is written in terms of the inner product with (17) as

[TABLE]

From the master equations (18) and Lemma 2.1, the time-development of

[TABLE]

is described by

[TABLE]

Equivalently, it follows from (19) that

[TABLE]

Remark 3.1.

Combining the sum over vertexes at fixed time $t$ ,

[TABLE]

the identity

[TABLE]

and (23), one verifies that the derivative of $\left\langle\,{1_{\,V}}\,\right\rangle_{V}=\mathbb{E}_{\,p_{t}}\,[\,1_{\,V}\,]$ with respect to time vanishes:

[TABLE]

where $\dot{p}_{\,t}$ denotes derivative of $p_{\,t}$ with respect to time $t$ , $\dot{p}_{\,t}=\mathrm{d}p_{\,t}/\mathrm{d}t$ .

Remark 3.2.

For ${\cal O}^{\,\prime}={\cal O}+c\,1_{\,V}\in\Lambda^{\,0}(G)$ with $c\in\mathbb{R}$ being constant, one has

[TABLE]

The inner product for ${\cal O}_{\,A}^{\,1}\in\Lambda^{\,1}(G)$ and ${\cal O}_{\,B}^{\,1}\in\Lambda^{\,1}(G)$ is $\left\langle\,{\cal O}_{\,A}^{\,1}\,,\,{\cal O}_{\,B}^{\,1}\,\right\rangle_{\,E}$ . In the case where ${\cal O}_{\,B}^{\,1}=\mathrm{d}\dot{p}_{\,t}$ and ${\cal O}_{\,A}^{\,1}$ that does not depend on time $t$ , one uses (20) to obtain

[TABLE]

3.2 Detailed balance conditions

Let $p^{\,\mathrm{eq}}$ be a prescribed probability distribution function so that

[TABLE]

Impose for any connected states $x$ and $y$

[TABLE]

which are known as the detailed balance conditions [4, 11]. These conditions imply the following. When $p^{\,\mathrm{eq}}$ is a stationary solution of the master equations, $\mathrm{d}p^{\,\mathrm{eq}}(x)/\mathrm{d}t=0$ , one has from (13) that

[TABLE]

If (24) holds, then (25) is satisfied. In this subsection (24) is assumed to hold.

These conditions are written in the graph theoretic language as

[TABLE]

and

[TABLE]

The later leads to

[TABLE]

if $p^{\,\mathrm{eq}}(x)$ does not vanish for all $x\in V$ . Thus, $\pi(e)=1$ for any loop edge $e\in E$ , and

[TABLE]

To seek a reversible measure incorporating (26), let $m_{\,E}\in S^{\,1}(G)$ be such that

[TABLE]

from which $m_{\,E}(e)=m_{\,E}(\,\overline{e}\,)>0$ . Thus, this $m_{\,E}\in S^{\,1}(G)$ can be used for a reversible measure defining (2) if $m_{\,E}$ does not vanish.

The master equations (15) under the detailed balance conditions are written as

[TABLE]

This can be written in terms of $\mathrm{d}_{\pi}$ , that is (11) with $\varphi=\pi$ , as

[TABLE]

The above equations involve $\mathrm{d}_{\,\pi}$ . In (28), there is no loop edge contribution since $\pi(e)=1$ and $(\mathrm{d}_{\,\pi}\,p_{\,t})(e)=0$ for any loop edge $e\in E$ , (See Remark 2.2). Although there is no self-adjoint operator in (28), there exists a way to express master equations in terms of a Laplacian, which is accomplished by a change of variables.

The following is the main theorem in this paper:

Theorem 3.2.

(Diffusion equations from master equations). Consider the master equation (15) in the case that

the prescribed stationary solution does not vanish, that is, $p^{\,\mathrm{eq}}(x)\neq 0$ , for all $x\in V$ , and 2. 2.

the detailed balance conditions are satisfied, that is, $w(e)$ satisfies (26).

Choose the measures $m_{\,V}\in\Lambda^{\,0}(G)$ and $m_{\,E}\in S^{\,1}(E)$ to be

[TABLE]

respectively. Then introduce $\psi_{\,t}\in\Lambda^{\,0}(G)$ such that

[TABLE]

where $p_{\,t}$ is a solution to the master equation (15). This function $\psi_{\,t}$ satisfies the diffusion equation

[TABLE]

where $\Delta_{\,V}$ has been defined in (7).

Proof.

Substituting (30) into (27), and using the definition of the Laplacian (7), one can complete the proof. The details are as follows:

From (30) and (27), and $\mathrm{d}p^{\,\mathrm{eq}}(x)/\mathrm{d}t=0$ , one has

[TABLE]

from which

[TABLE]

Since the action of the Laplacian (8) with (29) on $\psi_{\,t}$ is

[TABLE]

one obtains (31). ∎

Remark 3.3.

The explicit expression (32) reduces further. It follows from ${\mathrm{o}}({e})=x$ for $e\in E_{\,x}$ that

[TABLE]

Remark 3.4.

The conservation law of the sum

[TABLE]

can be shown below. With (31), one can show

[TABLE]

Remark 3.5.

Theorem 3.2 and (8) indicate that (31) does not involve any loop edge. This property also holds for the equations written in terms of $p_{\,t}(x)$ , (27).

Remark 3.6.

Since the diffusion equation (31) can be written as

[TABLE]

one has the form of continuity equation by introducing $\Pi_{\,t}\in\Lambda^{\,1}(G)$

[TABLE]

where $\mathrm{div}:\Lambda^{\,1}(G)\to\Lambda^{\,0}(G)$ has been defined in (6). For the case where the detailed balance conditions need not hold, but the conditions $w(\overline{e})=w(e)$ hold for all $e\in E$ , Fick’s law (16) holds. Besides, in the present case, the corresponding Fick’s law is

[TABLE]

for all $e\in E$ . Hence the current on $e\in E$ is expressed as the difference between $\psi_{\,t}({\mathrm{t}}({e}))$ and $\psi_{\,t}({\mathrm{o}}({e}))$ as follows:

[TABLE]

The following is an example of how this formulation can be applied to a model studied in nonequilibrium statistical mechanics:

Example 3.3.

(Kinetic Ising model without spin-coupling, [25]). Let $\sigma=\pm 1$ be a spin variable, and $p_{\,\mathrm{Ising}}^{\,\mathrm{eq}}(\sigma)$ an equilibrium distribution of $\sigma$ . Consider the master equations

[TABLE]

where the detailed balance condition is satisfied:

[TABLE]

This set of master equations induces $G=(V,E)$ and $w\in C^{\,1}(G)$ . This $G$ consists of $V=\{1,-1\}$ and $E=\{e^{\,\prime},e^{\,\prime\prime}\}$ , where $E_{\,1}=\{e^{\,\prime}\}$ and $E_{\,-\,1}=\{e^{\,\prime\prime}\}$ . The $G$ and $w\in C^{\,1}(G)$ are such that

[TABLE]

Introduce $\psi_{\,t,\mathrm{Ising}}$ such that

[TABLE]

With these variables and the detailed balance condition

[TABLE]

one has that

[TABLE]

from which

[TABLE]

This derived set of equations is consistent with (33).

Note that diffusion equations in Example 3.2 are not obtained by applying Theorem 3.2, since the detailed balance conditions are not satisfied.

To investigate the long-time behavior of the system (31), rewrite (31) as

[TABLE]

where ${\cal F}:\Lambda^{\,0}(G)\to\Lambda^{\,0}(G)$ has been defined such that

[TABLE]

The following Lemmas will be used:

Lemma 3.1.

The non-trivial solutions, $\psi^{\,(0)}(x)\neq 0$ , to the equations $(\Delta_{\,V}\psi^{\,(0)})(x)=0$ for any $x\in V$ are $\psi^{\,(0)}(x)=\psi_{\,0}^{\,(0)}$ , where $\psi_{\,0}^{\,(0)}\in\mathbb{R}$ is constant.

Proof.

For general $\psi$ , it follows that

[TABLE]

The equality holds only when $\mathrm{d}\psi=0$ . It implies that

[TABLE]

Since

[TABLE]

and the assumption that the graph is connected, the solution is $\psi^{\,(0)}(x)=\psi_{\,0}^{\,(0)}$ with $\psi_{\,0}^{\,(0)}$ being constant. ∎

Lemma 3.1 states that the steady state $\psi^{\,(0)}$ for (31), $\mathrm{d}\psi^{\,(0)}/\mathrm{d}t=0$ , is $\psi^{\,(0)}=\psi_{\,0}^{\,(0)}1_{\,V}$ . In other words, this $\psi_{\,0}^{\,(0)}1_{\,V}$ forms a fixed point set for this system. The following describes the stability for $\psi=\psi_{\,0}^{\,(0)}1_{\,V}$ :

Lemma 3.2.

The state $\psi_{\,t}=\psi_{\,0}^{\,(0)}1_{\,V}$ is asymptotically stable for (37), where $\psi_{\,0}^{\,(0)}\in\mathbb{R}$ is constant.

Proof.

Introduce the dynamical system for $\widetilde{\psi}\in\Lambda^{\,0}(G)$ ,

[TABLE]

where $\widetilde{{\cal F}}:\Lambda^{\,0}(G)\to\Lambda^{\,0}(G)$ has been defined such that

[TABLE]

It immediately follows that

[TABLE]

where $0_{\,V}$ is the element of $\Lambda^{\,0}(G)$ such that $0_{\,V}(x)=0$ for any $x\in V$ . This states that $0_{\,V}\in\Lambda^{\,0}(G)$ is a fixed point set for (38). The relation between (38) and (37) is as follows. If $\psi_{\,t}$ is a solution to (37), then $\widetilde{\psi}_{\,t}=\psi_{\,t}-\psi_{\,0}^{\,(0)}\,1_{\,V}$ is a solution to (38). This is due to

[TABLE]

On the contrary, if $\widetilde{\psi}_{\,t}$ is a solution to (38), then $\psi_{\,t}=\widetilde{\psi}_{\,t}+\psi_{\,0}^{\,(0)}\,1_{\,V}$ is a solution to (37). This is due to

[TABLE]

Hence, the solution to (37) is obtained by $\psi_{\,t}=\widetilde{\psi}_{\,t}+\psi_{\,0}^{\,(0)}1_{\,V}$ with $\widetilde{\psi}_{\,t}$ being a solution to (38).

Then define ${\cal L}:\Lambda^{\,0}(G)\to\mathbb{R}$ such that

[TABLE]

It immediately follows that ${\cal L}(\,\widetilde{\psi}_{\,t}\,)\geq 0$ for any $t\in\mathbb{R}$ , where $\widetilde{\psi}_{\,t}$ is a solution to (38). In addition, one has that $\mathrm{d}{\cal L}(\widetilde{\psi}_{\,t})/\mathrm{d}t\leq 0$ for any $t\in\mathbb{R}$ , since

[TABLE]

From these properties, ${\cal L}$ is a Lyapunov function. Applying these statements to the Lyapunov stability theorem, one has that $\widetilde{\psi}_{\,t}=0_{\,V}$ in the dynamical system (38) is asymptotically stable. This yields that $\psi_{\,0}^{\,(0)}1_{\,V}$ is asymptotically stable for (37). ∎

Then one has the following from Theorem 3.2:

Corollary 3.1.

(Convergence of solutions to master equations).

[TABLE]

Proof.

It follows from Lemmas 3.1 and 3.2 that

[TABLE]

This and the normalization condition $\sum_{x\in V}p^{\,\mathrm{eq}}(x)=1$ lead to

[TABLE]

Meanwhile, the conservation law (34) holds. Thus, the left hand side of the equation above is unity:

[TABLE]

Therefore,

[TABLE]

∎

It has been shown that discrete diffusion equations are derived from master equations under the conditions that the detailed balance conditions are satisfied. Its converse statement under these conditions also holds.

Proposition 3.1.

*(From diffusion equations to master equations). Let $w\in C^{\,1}(G)$ be a transition matrix, and $p^{\,\mathrm{eq}}\in\Lambda^{\,0}(G)$ an equilibrium distribution function. Assume that the detailed balance conditions (26) hold. Choose $m_{\,V}\in\Lambda^{\,0}(G)$ and $m_{\,E}\in S^{\,1}(G)$ so that $m_{\,V}(x)=p^{\,\mathrm{eq}}(x)$ and $m_{\,E}(e)=w(e)p^{\,\mathrm{eq}}({\mathrm{o}}({e}))$ as in (29). Then, the diffusion equations (31) yield master equations. *

Proof.

Introduce the inner products (1) and (2). In addition, introduce $p_{\,t}\in\Lambda^{\,0}(G)$ that depends on $t\in\mathbb{R}$ such that $p_{\,t}(x)=p^{\,\mathrm{eq}}(x)\psi_{\,t}(x)$ . Then multiplying both sides of the diffusion equations by $p^{\,\mathrm{eq}}(x)$ , one has

[TABLE]

With (26), the equations above can be written as

[TABLE]

This can also be written in terms of $p_{\,t}(x)=p^{\,\mathrm{eq}}(x)\psi_{\,t}(x)$ as

[TABLE]

∎

So far, basic statements of the system have been given. These include asymptotic behavior of the diffusion equations, Lemma 3.2. Furthermore, if the eigenvalues and eigenfunctions of the Laplacian $\Delta_{\,V}$ are known, then an explicit solution for the initial value problem of the diffusion equations (31) can be obtained. Since the diffusion equations are linear, it is enough to consider the case where the solution space is a linear space of $\Lambda^{\,0}(G)$ . Moreover, from Theorem 3.2, one has the equation on $\Lambda^{\,1}(G)$

[TABLE]

Since (39) is linear, eigenvalues and eigenfunctions of $\Delta_{\,E}$ play roles. Thus in what follows $\Lambda^{\,0}(G)$ and $\Lambda^{\,1}(G)$ denote linear spaces. Then how to construct an explicit solution to the diffusion equations is argued below.

Let $A$ be a linear operator acting on a finite-dimensional vector space ${\cal U}$ , and $\lambda_{\,A}$ a non-zero number satisfying

[TABLE]

where $\phi\in{\cal U}$ is not [math]. Then $\lambda_{\,A}$ and $\phi$ are referred to as an non-zero eigenvalue and its associated or corresponding eigenfunction, respectively. For a self-adjoint operator $A$ , it is known that all the eigenvalues are real, $\lambda_{\,A}\in\mathbb{R}$ . In addition, if all the non-zero eigenvalues are positive, then $A$ is referred to as a positive operator. If $A\,\phi=0$ with non-zero $\phi$ , then $\phi$ is referred to as the eigenfunction associated with the zero-eigenvalue.

Since the Laplacian $\Delta_{\,V}$ is a linear operator acting on $\Lambda^{\,0}(G)$ and is self-adjoint as shown in Lemma 2.2, one can discuss if $-\Delta_{\,V}$ is a positive operator or not. Then one has the following:

Lemma 3.3.

The negative of the Laplacian acting on $\Lambda^{\,0}(G)$ , $-\Delta_{\,V}$ , is a positive operator.

Proof.

Let $\widetilde{\lambda}_{\,V}^{\,(s)}$ be a non-zero eigenvalue labeled by $s$ , and $\widetilde{\phi}_{\,V}^{\,(s)}$ an associated eigenfunction of $-\Delta_{\,V}$ so that $-\Delta_{\,V}\widetilde{\phi}_{\,V}^{\,(s)}=\widetilde{\lambda}_{\,V}^{\,(s)}\widetilde{\phi}_{\,V}^{\,(s)}$ . Then it follows that

[TABLE]

for all $\widetilde{\phi}_{\,V}^{\,(s)}\neq 0_{\,V}$ . Applying the assumed conditions $\left\langle\,\widetilde{\phi}_{\,V}^{\,(s)}\,,\,\widetilde{\phi}_{\,V}^{\,(s)}\,\right\rangle_{\,V}>0$ and $\widetilde{\lambda}_{\,V}^{\,(s)}\neq 0$ to the obtained inequality $\widetilde{\lambda}_{\,V}^{\,(s)}\left\langle\,\widetilde{\phi}_{\,V}^{\,(s)}\,,\,\widetilde{\phi}_{\,V}^{\,(s)}\,\right\rangle_{\,V}\geq\,0$ , one concludes that $\widetilde{\lambda}_{\,V}^{\,(s)}>0$ . ∎

Remark 3.7.

Repeating similar arguments in the proof of Lemma 3.3, one has that all the eigenvalues of $\Delta_{\,V}$ are negative or zero,

[TABLE]

where $\phi_{\,V}^{\,(s)}\in\Lambda^{\,0}(G)$ is the eigenfunction associated with $\lambda_{\,V}^{\,(s)}$ for $s\neq 0$ . For $s=0$ , $\phi_{\,V}^{\,(0)}$ denotes the eigenfunction associated with the zero-eigenvalue $\lambda_{\,V}^{\,(0)}=0$ .

Moreover, given a self-adjoint operator $A$ acting on a linear space ${\cal U}$ with an inner product $\left\langle\,\,,\,\,\right\rangle_{\,{\cal U}}$ , it is known that ${\cal U}$ has an orthonormal basis consisting of eigenvectors of $A$ . By applying this, one has the decomposition

[TABLE]

where $s$ denotes a label for eigenfunction, ${\cal N}_{\,V}$ the totality of labels for non-zero eigenfunctions, $a^{\,(s)}$ the real-valued function labeled by $s$ of $t$ , and $\delta^{\,ss^{\,\prime}}$ the Kronecker delta, giving unity if $s=s^{\,\prime}$ and zero otherwise.

The normalized eigenfunction associated with the zero-eigenvalue $\phi_{\,V}^{\,(0)}$ is obtained as follows:

Lemma 3.4.

The normalized zero-eigenfunction is $\phi_{\,V}^{\,(0)}=1_{\,V}$ :

[TABLE]

Proof.

It follows that $\Delta_{\,V}1_{\,V}=-\mathrm{d}^{\,\dagger}\mathrm{d}1_{\,V}=0$ . Then, the normalization condition is verified as

[TABLE]

∎

To discuss properties of the zero-eigenfunction further, let $\ker\Delta_{\,V}$ be the space spanned by the zero-eigenfunction $\phi_{\,V}^{\,(0)}$ :

[TABLE]

Then, one has the following:

Lemma 3.5.

A normalized zero-eigenfunction in Lemma 3.4 is unique up to sign, and $\dim(\ker\Delta_{\,V}^{\,(0)})=1$ .

Proof.

First, $\dim(\ker\Delta_{\,V}^{\,(0)})=1$ is proved. The proof below is similar to the proof of Lemma 3.1.

For general $\psi\in\Lambda^{\,0}(G)$ , it follows that

[TABLE]

The equality holds only when $\mathrm{d}\,\psi=0$ . It implies for $\phi_{\,V}^{\,(0)}\in\ker\Delta_{\,V}$ that

[TABLE]

Thus, for all $e\in E$ ,

[TABLE]

or equivalently,

[TABLE]

Combining this equality and the assumption that the graph $G$ is connected, one has that

[TABLE]

where $\phi_{\,V,0}^{\,(0)}\in\mathbb{R}$ is constant. Thus $\ker\Delta_{\,V}^{\,(0)}=\mathrm{span}\,\{\,1_{\,V}\}$ , and the equality $\dim(\ker\Delta_{\,V}^{\,(0)})=1$ holds.

In general, given a non-zero vector, the normalized vector is uniquely determined up to sign. It follows from this uniqueness with $\dim(\ker\Delta_{\,V}^{\,(0)})=1$ that the normalized zero-eigenfunction is uniquely determined up to sign. ∎

The sign ambiguity in Lemma 3.5 is fixed when one constructs $p_{\,t}(x)=p^{\,\mathrm{eq}}(x)\,\psi_{\,t}(x)$ with the properties $p^{\,\mathrm{eq}}(x)>0$ and $\psi_{\,t}(x)=\sum_{s}a^{\,(s)}(t)\phi_{\,V}^{\,(s)}(x)\geq 0$ for all $t\in\mathbb{R}$ and $x\in V$ . Observe from Lemma 3.5 that the number of the elements of ${\cal N}_{\,V}$ is calculated as $|\,{\cal N}_{\,V}\,|=\#V-1$ .

With these Lemmas, the solution to (31), $\psi_{\,t}$ , is obtained as follows:

Proposition 3.2.

(Spectrum decomposition). Let $\{\lambda_{\,V}^{\,(s)}\}$ be the totality of non-zero eigenvalues of $\Delta_{\,V}$ , and $\{\phi_{\,V}^{\,(s)}\}$ that of the corresponding eigenfunctions. Then

[TABLE]

*is a solution to the diffusion equations (31) derived from the master equations. *

Proof.

From (40), it follows that

[TABLE]

With these equations and (31), one has

[TABLE]

from which $a^{\,(s)}(t)=a^{\,(s)}(0)\exp(\lambda_{\,V}^{\,(s)}t)$ . Taking into account Remark 3.7, one can write $\lambda_{\,V}^{\,(s)}=-|\lambda_{\,V}^{\,(s)}|\leq 0$ . Combining these arguments, one has

[TABLE]

Finally it follows from Corollary 3.1 that $a^{\,(0)}(0)$ above is unity. This completes the proof. ∎

Notice that

[TABLE]

and

[TABLE]

An example of the spectrum decomposition of $\psi_{\,t}$ is given below.

Example 3.4.

(Eigen system for kinetic Ising model without spin-coupling). Consider Example 3.3. The matrix form of (36) is immediately obtained as

[TABLE]

Two eigenvalues of $M_{\,\mathrm{Ising}}$ are obtained as

[TABLE]

Their orthonormal eigenfunctions are

[TABLE]

where

[TABLE]

The orthonormality is verified as

[TABLE]

where the normalization of $p_{\,\mathrm{Ising}}^{\,\mathrm{eq}}$ and the detailed balance condition have been used. In Ref. [25], the equilibrium distribution $p_{\,\mathrm{Ising}}^{\,\mathrm{eq}}$ and $w_{\,\sigma\to-\sigma}$ were chosen so that they depend on parameters $\theta\in\mathbb{R}$ and $\gamma\in\mathbb{R}_{\,+}$ as

[TABLE]

The physical meaning of $\theta$ is a quantity that is proportional to the inverse temperature, and that of $\gamma$ is a characteristic relaxation time. The non-zero eigenvalue and its corresponding eigenfunction for this case are

[TABLE]

Thus, the spectrum decomposition (41) is obtained as

[TABLE]

where

[TABLE]

with $a_{\,\mathrm{Ising}}^{\,(1)}(0)\geq 0$ being an initial constant for the dynamical system $\dot{a}_{\,\mathrm{Ising}}^{\,(1)}=-\,\gamma\,a_{\,\mathrm{Ising}}^{\,(1)}$ .

So far eigenvalues and eigenfunctions of $\Delta_{\,V}$ have been investigated. There exists a link between eigenvalues of $\Delta_{\,V}$ and those of $\Delta_{\,E}$ . To state this, notations are introduced as follows. Let ${\cal N}_{\,E}$ be the totality of labels for non-zero eigenfunctions of $\Delta_{\,E}$ , $\mathrm{Spec}^{\,\prime}\Delta_{\,V}$ and $\mathrm{Spec}^{\,\prime}\Delta_{\,E}$ the totalities of non-zero eigenvalues of $\Delta_{\,V}$ and that of $\Delta_{\,E}$ , respectively. Moreover, recall $\ker\Delta_{\,V}$ , $\dim(\ker\Delta_{\,V})=1$ from Lemma 3.5, and let

[TABLE]

First, one has the following:

Lemma 3.6.

[TABLE]

Proof.

This proof can be split into two steps. First, $\ker\mathrm{d}\subseteq\ker\Delta_{\,V}$ is shown. Then, $\ker\Delta_{\,V}\subseteq\ker\mathrm{d}$ is shown. From these, one completes the proof.

(Proof of $\ker\mathrm{d}\subseteq\ker\Delta_{\,V}$ ): Take $\phi_{\,V}^{\,(0)}\in\ker\mathrm{d}$ , i.e., $\mathrm{d}\phi_{\,V}^{\,(0)}=0$ . Then it follows that

[TABLE]

from which $\ker\mathrm{d}\subseteq\ker\Delta_{\,V}$ .

(Proof of $\ker\Delta_{\,V}\subseteq\ker\mathrm{d}$ ): Take $\phi_{\,V}^{\,(0)}\in\ker\Delta_{\,V}$ , i.e., $\Delta_{\,V}\phi_{\,V}^{\,(0)}=0$ . Then it follows from

[TABLE]

that $\mathrm{d}\phi_{\,V}^{\,(0)}=0$ . Thus, $\ker\Delta_{\,V}\subseteq\ker\mathrm{d}$ .

These two steps yield $\ker\mathrm{d}=\ker\Delta_{\,V}$ . ∎

Similar to Lemma 3.6, one has the following:

Lemma 3.7.

[TABLE]

Proof.

A way to prove this is analogous to that of Lemma 3.6. ∎

Then one has the following property, referred to as supersymmetry [21]:

Theorem 3.3.

(Supersymmetry of Laplacians).

[TABLE]

Proof.

This proof can be split into two steps. First, $\mathrm{Spec}^{\,\prime}\Delta_{\,V}\subseteq\mathrm{Spec}^{\,\prime}\Delta_{\,E}$ is shown. Then, $\mathrm{Spec}^{\,\prime}\Delta_{\,E}\subseteq\mathrm{Spec}^{\,\prime}\Delta_{\,V}$ is shown. From these, one completes the proof.

( Proof of $\mathrm{Spec}^{\,\prime}\Delta_{\,V}\subseteq\mathrm{Spec}^{\,\prime}\Delta_{\,E}$ ): Assume that $\Delta_{\,V}\phi_{\,V}^{\,(s)}=\lambda_{\,V}^{\,(s)}\phi_{\,V}^{\,(s)}$ , $(s\in{\cal N}_{\,V})$ . Then, $\{\lambda_{\,V}^{\,(s)}\}=\mathrm{Spec}^{\,\prime}\Delta_{\,V}$ . From this assumption and Lemma 3.6, it follows that $\mathrm{d}\,\phi_{\,V}^{\,(s)}\neq 0$ . Since

[TABLE]

one has that $\lambda_{\,V}^{\,(s)}\in\mathrm{Spec}^{\,\prime}\Delta_{\,E}$ for each $s\in{\cal N}_{\,V}$ . This yields that $\mathrm{Spec}^{\,\prime}\Delta_{\,V}\subseteq\mathrm{Spec}^{\,\prime}\Delta_{\,E}$ .

( Proof of $\mathrm{Spec}^{\,\prime}\Delta_{\,E}\subseteq\mathrm{Spec}^{\,\prime}\Delta_{\,V}$ ): Assume that $\Delta_{\,E}\,\phi_{\,E}^{\,(s)}=\lambda_{\,E}^{\,(s)}\phi_{\,E}^{\,(s)}$ , $(s\in{\cal N}_{\,E})$ . Then, $\{\lambda_{\,E}^{\,(s)}\}=\mathrm{Spec}^{\,\prime}\Delta_{\,E}$ . From this assumption and Lemma 3.7, it follows that $\mathrm{d}^{\,\dagger}\,\phi_{\,E}^{\,(s)}\neq 0$ . Since

[TABLE]

one has that $\lambda_{\,E}^{\,(s)}\in\mathrm{Spec}^{\,\prime}\Delta_{\,V}$ for each $s\in{\cal N}_{\,E}$ . This yields that $\mathrm{Spec}^{\,\prime}\Delta_{\,E}\subseteq\mathrm{Spec}^{\,\prime}\Delta_{\,V}$ .

These two steps yield the desired equality. ∎

Due to Theorem 3.3, non-zero eigenvalues $\lambda_{\,V}^{\,(s)},(s\in{\cal N}_{\,V})$ and $\lambda_{\,E}^{\,(s)},(s\in{\cal N}_{\,E})$ are not distinguished. In what follows they are denoted $\lambda^{\,(s)}$ .

3.2.1 Expectation values

For the case where the detailed balance conditions are not imposed, how expectation values are described has been argued in Section 3.1.1. It is also of interest to formulate how expectation values are described for the case that the detailed balance conditions are satisfied. To establish such a formulation, a relation between an expectation value and the inner product on $\Lambda^{\,0}(G)$ is shown first. Second, inequalities for sums involving $\psi_{\,t}$ over vertexes are shown. Then some identities for such sums are shown. Finally dynamical systems for expectation variables are constructed. In general, one of the most fundamental roles in statistical mechanics is to link properties of macroscopic quantities and microscopic dynamical systems. For the case that macroscopic quantities can be identified with expectation variables, and microscopic dynamical variables can be identified with master equations, the approaches shown below are expected to be employed in a wider class of systems.

The expectation value of ${\cal O}^{\,0}\in\Lambda^{\,0}(G)$ is denoted by $\mathbb{E}_{\,p_{t}}[\,{\cal O}^{\,0}\,]$ as in (22). This expectation value is written in terms of the inner product with (29) as

[TABLE]

where $p_{\,t}(x)=p^{\,\mathrm{eq}}(x)\psi_{\,t}(x)$ in (30) has been used. The time-development of

[TABLE]

for ${\cal O}^{\,0}\in\Lambda^{\,0}(G)$ is then described by

[TABLE]

Equivalently, it follows from (35) that

[TABLE]

The H function or KL-divergence defined by

[TABLE]

plays a central role in information theory. It is known that [28]

[TABLE]

This can be proven in the present formulation as follows. The derivative of $D(p_{\,t}\|p^{\,\mathrm{eq}})$ with respect to $t$ is written in terms of $p^{\,\mathrm{eq}}(x)$ and $\psi_{\,t}(x)$ as

[TABLE]

Then applying the inequality

[TABLE]

and (31), one has

[TABLE]

Finally from $\left\langle\,\mathrm{d}\psi_{\,t}\,,\,\mathrm{d}\psi_{\,t}\,\right\rangle_{V}\geq 0$ , one has the desired inequality:

[TABLE]

It is then of interest to explore similar inequalities for sums involving $\psi_{\,t}$ over vertexes. Define $S_{\,t}\in\Lambda^{\,0}(G)$ to be

[TABLE]

A significance of $S_{\,t}$ is discussed after showing the following Proposition:

Proposition 3.3.

(Inequality for $S_{\,t}$ ).

[TABLE]

Proof.

By applying the inequality

[TABLE]

to the equality

[TABLE]

one has

[TABLE]

∎

From (44), the quantity $S_{\,t}$ obeys an integral fluctuation theorem in the sense that [26],

[TABLE]

Fluctuation theorems hold in far from equilibrium states, and thus they are used as basic tools in the study of nonequilibrium statistical mechanics [27]. Note that Proposition 3.3 can be written as $\mathbb{E}_{\,p^{\,\mathrm{eq}}}[\,S_{\,t}\,]\geq 0$ . Related to this, it is shown below how the time-dependence of $\mathbb{E}_{\,p_{t}}[\,S_{\,t}\,]$ is related to eigenvalues of the derived diffusion equations. To this end, define

[TABLE]

Then one has the following.

Proposition 3.4.

(Inequality for time-derivative of $H[\psi_{\,t}]$ ).

[TABLE]

Proof.

Substituting the inequality

[TABLE]

and (31) into (45), one has

[TABLE]

From this and (40), it follows that

[TABLE]

from which

[TABLE]

Since $\lambda^{(s)}=-|\lambda^{(s)}|<0$ for $s\in{\cal N}_{\,V}$ (See Remark 3.7), one has that

[TABLE]

∎

To construct a dynamical system for an expectation variable $\left\langle\,{{\cal O}^{\,0}}\,\right\rangle_{V}=\mathbb{E}_{\,p_{t}}[\,{\cal O}^{\,0}\,]$ with the spectrum decomposition, one starts with

[TABLE]

where (43) has been used. The right hand side of the equation above might not be written in terms of a function of

[TABLE]

For this reason, it is not obvious whether there exists a closed dynamical system for $\left\langle\,{{\cal O}^{\,0}}\,\right\rangle_{V}$ . Nevertheless, for some simple systems, such closed dynamical systems can be constructed from master equations. For example, one has the following:

Proposition 3.5.

(Dynamical system for master equations with a unique non-trivial eigenvalue). Consider the case where $|\,{\cal N}_{\,V}\,|=1$ . In this case, one has the dynamical system on $\mathbb{R}$

[TABLE]

Proof.

Since $|\,{\cal N}_{\,V}\,|=1$ , one has from (46) that

[TABLE]

and from (47) that

[TABLE]

Substituting the second equation into the first one, one completes the proof. ∎

The solution to (48) is obtained as

[TABLE]

It follows from this expression with $\lambda^{\,(1)}<0$ that the equilibrium state for $\left\langle\,{{\cal O}^{\,0}}\,\right\rangle_{V}$ is indeed uniquely realized as $t\to\infty$ :

[TABLE]

The following is an example associated with Proposition 3.5:

Example 3.5.

(Dynamical system for magnetization driven by kinetic Ising model). Consider Examples 3.3 and 3.4. In this analysis $p_{\,\mathrm{Ising}}^{\,\mathrm{eq}}(\sigma;\theta)$ is abbreviated as $p_{\,\mathrm{Ising}}^{\,\mathrm{eq}}(\sigma)$ . Choose ${\cal O}^{\,0}\in\Lambda^{\,0}(G)$ to be

[TABLE]

The expectation variable of this ${\cal O}^{\,0}$ is time-dependent magnetization, expressed as

[TABLE]

Its derivative with respect to $t$ is calculated, and that derivative can be expressed in terms of $\left\langle\,{\sigma}\,\right\rangle_{V}$ as

[TABLE]

Substituting $p_{\,\mathrm{Ising}}^{\,\mathrm{eq}}(\sigma)=\mathrm{e}^{\,\theta\,\sigma}/(2\cosh\theta)$ into the equation above, one has the closed dynamical system for $\left\langle\,{\sigma}\,\right\rangle_{V}$

[TABLE]

which is a dynamical system studied in Ref. [25].

There exist closed dynamical systems for expectation variables $\mathbb{E}_{\,p_{t}}[\,{\cal O}^{\,0}\,]$ with some appropriately chosen ${\cal O}^{\,0}$ . Such systems are given as follows:

Proposition 3.6.

(Dynamical systems for expectation variables 1). Consider the case where

[TABLE]

Then,

[TABLE]

follows the dynamical system

[TABLE]

Proof.

It follows from (42) and (40) that

[TABLE]

From this and (42), one has the closed form

[TABLE]

∎

To show how (49) is linked to another dynamical system in another scientific literature, it is shown below that the diffusion equations derived from master equations (31) are used to facilitate the derivation of a dynamical system discussed in information geometry. Here information geometry is a geometrization of mathematical statistics [29].

In the context of dynamical systems theory in information geometry, the dynamical system

[TABLE]

has been studied in [30, 31, 32, 33]. The variables $\{\eta_{\,j}\}$ in (50) correspond to the expectation values for an exponential family of probability distribution functions, $\{\eta_{\,j}^{\,\prime}\}$ are constants, and $d$ is the number of canonical parameters of the exponential family. In Ref. [31], (50) was constructed as a gradient flow by introducing the so-called divergence function. By contrast, how to establish (50) in terms of probability distribution functions obeying master equations has remained unclear.

Introduce a dynamical system that is slightly different from (50)

[TABLE]

with some scaled time variables $t^{\,(j)}$ . It is shown below how (49) is linked to (51).

Proposition 3.7.

The dynamical system (49) yields (51).

Proof.

First, introduce $\widetilde{\eta}_{\,s}$ and $t^{\,(s)}$ so that

[TABLE]

from which

[TABLE]

Substituting this into (49), one has

[TABLE]

Second, introduce $\eta_{\,s}(\,t^{\,(s)}\,)$ and constant $\eta_{\,s}^{\,\prime}$ so that

[TABLE]

The variable $\eta_{\,s}$ obeys

[TABLE]

This is (51) with $s$ being replaced with $j$ . ∎

Similar to Proposition 3.6, closed equations on $\Lambda^{\,1}(G)$ are obtained as follows. Define for ${\cal O}^{\,1}\in\Lambda^{\,1}(G)$ ,

[TABLE]

Differentiating this equation with respect to $t$ and using (39), one has

[TABLE]

Then one has the following:

Proposition 3.8.

*(Dynamical systems for expectation variables 2). Consider the case where *

[TABLE]

Then, one has the closed dynamical system

[TABLE]

Proof.

It follows from that

[TABLE]

∎

4 Conclusions

This paper offers a viewpoint that master equations employed in nonequilibrium statistical mechanics can be analyzed in discrete geometry. As the main theorem in this paper, when the detailed balance conditions are satisfied, master equations induce diffusion equations, where Laplacians are associated with chosen inner products. These diffusion equations can analytically be solved and then explicit solutions have been obtained. With the self-adjoint property of the Laplacians, the orthonormal decomposition of the solution and closed equations for expectation variables have been obtained. Furthermore, the coboundary operator and its adjoint enable us to simplify various calculations including expectation values, and to show the so-called supersymmetry for two classes of Laplacians. Moreover, to illustrate how to apply these, some examples have been shown.

The significance of this discrete geometric description of master equations is argued from a mathematical viewpoint as follows. Although various graph theoretic approaches to master equations exist in the literature, none of them have yielded reduced equations such as diffusion equations without any approximation. A crucial point of the derivation of the diffusion equations is to introduce freedom to choose weights defining inner products for functions on graphs. Here the introduction of weights, or equivalently the additional freedom, has been incorporated in discrete geometry [16], and has then induced wider classes of Laplacians on graphs whose eigenvalues are real due to the self-adjoint property. The usefulness of this self-adjoint property of Laplacians has been employed throughout this paper.

The significance of the present study is argued from a physical viewpoint as follows. Given master equations satisfying the detailed balance conditions, diffusion equations have been obtained by introducing new variables. Since diffusion equations are ubiquitous in physics, insight can be gained from the physical literature. For instance, such insight is to identify the time-development of master equations with how an ink is spread in a fluid. It is one of relaxation processes in nonequilibrium systems, and this time-development of master equations can be expressed as the sum of exponential decays with real decay rates. Thus, oscillatory behavior in time is not observed in the new variables. Meanwhile, inner products have been introduced by choosing measures in this paper. Choosing these measures has provided preferred or biased ways to describe the time-development of the master equations. Hence, other choices of measures could be suitable for master equations with other conditions. In addition, the use of inner products is similar to that in quantum mechanics [7]. This similarity is expected to be useful for introducing notions developed in quantum mechanics to the study of master equations. It should be mentioned that one of the most fundamental roles in statistical mechanics is to link properties of macroscopic quantities and microscopic dynamical systems. When macroscopic quantities can be identified with expectation variables, and microscopic dynamical variables can be identified with master equations, the approaches to construct dynamical systems for expectation variables shown in this paper are expected to be used in a wider class of systems.

There are numbers of extensions that follow from this study. One of them is to apply the present approach to quantum systems, chemical reactions, and economical systems. Another example is to consider the case where the detailed balance conditions are not satisfied [10]. Moreover, some infinite graph case and discrete-time case should be addressed, and their applications to mathematical engineering, including Monte-Carlo simulations, should be considered as future works.

Acknowledgments

S.G. was partially supported by JSPS (KAKENHI) Grant No. JP19K03635 and is also grateful to Shuhei MANO for fruitful discussions. H.H. was partially supported by JSPS (KAKENHI) Grant No. JP17H01793.

Bibliography33

The reference list from the paper itself. Each links out to its DOI / PubMed record.

1[1] R. Kubo, et Al, Statistical physics II , Springer (1991).
2[2] J. Krafter and I.M. Sokolov, First Steps in Random Walks: From Tools to Applications , Oxford University Press, (2011).
3[3] M.F. Weber and E. Frey, Rep. Prog. Phys., 80 , 046601, (2017).
4[4] N.G. Van Kampen, Stochastic Processes in Physics and Chemistry , 3rd edition, North Holland, (2007).
5[5] K. Binder, Rep. Prog. Phys., 60 , 487–599, (1997).
6[6] G. Lindblad, Commun. math. Phys. 48 , 119–130, (1976).
7[7] J. Baez and J.D. Biamonte, Quantum Techniques in Stochastic Mechanics , World Scientific, (2018).
8[8] S. Goto and H. Hino, Geometric Science of Information. GSI 2019 , Springer Lectures Notes in Computer Science (Springer, 2019), 11712 , 239–247.

TL;DR

Contribution

Findings

Abstract

Peer Reviews

Videos

Diffusion equations from master equations

Abstract

1 Introduction

Claim*.*

Claim*.*

2 Preliminaries

2.1 Standard operators

Lemma 2.1**.**

Proof**.**

Remark 2.1**.**

Lemma 2.2**.**

Proof**.**

Lemma 2.3**.**

Proof**.**

2.2 Operators for master equations

Remark 2.2**.**

Lemma 2.4**.**

Proof**.**

3 Master equations

3.1 Graph formulation

Example 3.1**.**

Theorem 3.1**.**

Proof**.**

Example 3.2**.**

3.1.1 Expectation values

Remark 3.1**.**

Remark 3.2**.**

3.2 Detailed balance conditions

Theorem 3.2**.**

Proof**.**

Remark 3.3**.**

Remark 3.4**.**

Remark 3.5**.**

Remark 3.6**.**

Example 3.3**.**

Lemma 3.1**.**

Proof**.**

Lemma 3.2**.**

Proof**.**

Corollary 3.1**.**

Proof**.**

Proposition 3.1**.**

Proof**.**

Lemma 3.3**.**

Proof**.**

Remark 3.7**.**

Lemma 3.4**.**

Proof**.**

Lemma 3.5**.**

Proof**.**

Proposition 3.2**.**

Proof**.**

Example 3.4**.**

Lemma 3.6**.**

Proof**.**

Lemma 3.7**.**

Proof**.**

Theorem 3.3**.**

Proof**.**

3.2.1 Expectation values

Proposition 3.3**.**

Proof**.**

Proposition 3.4**.**

Proof**.**

Proposition 3.5**.**

Proof**.**

Example 3.5**.**

Proposition 3.6**.**

Proof**.**

Proposition 3.7**.**

Proof**.**

Proposition 3.8**.**

Proof**.**

4 Conclusions

*Claim**.*

*Claim**.*

Lemma 2.1.

Proof.

Remark 2.1.

Lemma 2.2.

Proof.

Lemma 2.3.

Proof.

Remark 2.2.

Lemma 2.4.

Proof.

Example 3.1.

Theorem 3.1.

Proof.

Example 3.2.

Remark 3.1.

Remark 3.2.

Theorem 3.2.

Proof.

Remark 3.3.

Remark 3.4.

Remark 3.5.

Remark 3.6.

Example 3.3.

Lemma 3.1.

Proof.

Lemma 3.2.

Proof.

Corollary 3.1.

Proof.

Proposition 3.1.

Proof.

Lemma 3.3.

Proof.

Remark 3.7.

Lemma 3.4.

Proof.

Lemma 3.5.

Proof.

Proposition 3.2.

Proof.

Example 3.4.

Lemma 3.6.

Proof.

Lemma 3.7.

Proof.

Theorem 3.3.

Proof.

Proposition 3.3.

Proof.

Proposition 3.4.

Proof.

Proposition 3.5.

Proof.

Example 3.5.

Proposition 3.6.

Proof.

Proposition 3.7.

Proof.

Proposition 3.8.

Proof.