Absence of correlations in the energy exchanges of an exactly solvable model of heat transport with many degrees of freedom

TL;DR

This paper analyzes an exactly solvable heat conduction model showing that, with many degrees of freedom, energy exchange correlations vanish and the system's evolution simplifies to the classical heat equation.

Contribution

It demonstrates that in a generalized Kipnis--Marchioro--Presutti model, correlations disappear as degrees of freedom increase, reducing the dynamics to local temperature evolution.

Findings

Heat conductivity is proportional to the interaction rate.

Correlations between energy variables vanish with many degrees of freedom.

The model reduces to the discrete heat equation in the large degrees of freedom limit.

Abstract

A process based on the exactly solvable Kipnis--Marchioro--Presutti model of heat conduction [J. Stat. Phys. 27 65 (1982)] is described whereby lattice cells share their energies among many identical degrees of freedom while, in each cell, only two of them are associated with energy exchanges connecting neighbouring cells. It is shown that, up to dimensional constants, the heat conductivity is half the interaction rate, regardless of the degrees of freedom. Moreover, as this number becomes large, correlations between the energy variables involved in the exchanges vanish. In this regime, the process thus boils down to the time-evolution of the local temperatures which is prescribed by the discrete heat equation.

Equations

$$\begin{gathered} K({\xi }_a,{\xi }_b\to {\xi }_a-\eta ,{\xi }_b+\eta )={\int }_0^{{\xi }_a}\text{upd}{ϵ}_a{Q}_{{\nu }_a,{\xi }_a}({ϵ}_a) \\ \times {\int }_0^{{\xi }_b}\text{upd}{ϵ}_b{Q}_{{\nu }_b,{\xi }_b}({ϵ}_b)\frac{{\tau }^{-1}}{{ϵ}_a+{ϵ}_b}{\Theta }_{\text{textsc}h}({ϵ}_a-\eta ){\Theta }_{\text{textsc}h}({ϵ}_b+\eta ), \end{gathered}$$

$$Q_{\nu ,\xi }(ϵ)=(\nu -1){\xi }^{-1}(1-ϵ/\xi {)}^{\nu -2}{\Theta }_{\text{textsc}h}(\xi -ϵ){\Theta }_{\text{textsc}h}(ϵ).$$

$$P_{\text{textsc}eq}^{(N)}({\underline{\xi }}_N)K({\xi }_a,{\xi }_b\to {\xi }_a-\eta ,{\xi }_b+\eta )\text{cr}=P_{\text{textsc}eq}^{(N)}({\underline{\xi }}_N^{a,b,\eta })K({\xi }_a-\eta ,{\xi }_b+\eta \to {\xi }_a,{\xi }_b),$$

$$\nu_{\textsc{t}}^{1-\nu_{\textsc{t}}}(\nu_{\textsc{t}}-1)!\prod_{i}\frac{\beta^{\nu_{i}}\xi_{i}^{\nu_{i}-1}}{(\nu_{i}-1)!}\,\delta_{\textsc{d}}\!\Big{[}\sum_{i}\nolimits(\beta\,\xi_{i}-\nu_{i})\Big{]}\,.$$

$$p_{\nu ,\beta }(\xi )=\frac{{\beta }^{\nu }{\xi }^{\nu -1}}{(\nu -1)!}{\mathrm{e}}^{-\beta \xi }.$$

$$f_n({\xi }_a,{\xi }_b)$$

$$f_1({\xi }_a,{\xi }_b)=\frac{1}{2\tau }\left(\frac{{\xi }_a}{{\nu }_a}-\frac{{\xi }_b}{{\nu }_b}\right),$$

$$f_2({\xi }_a,{\xi }_b)=\frac{2}{3\tau }\left(\frac{{\xi }_a^2}{{\nu }_a({\nu }_a+1)}-\frac{{\xi }_a{\xi }_b}{2{\nu }_a{\nu }_b}+\frac{{\xi }_b^2}{{\nu }_b({\nu }_b+1)}\right).$$

$$p_{1,{\beta }_{\pm }}(ϵ)={\beta }_{\pm }{\mathrm{e}}^{-{\beta }_{\pm }ϵ},$$

$${\partial }_{t}P={\hat{\mathcal{L}}}^{†}P=\sum _{a={}^{-N}/{}_2}^{{}^N/{}_2-1}{\hat{\mathcal{L}}}_{a,a+1}^{†}P+{\hat{\mathcal{L}}}_{{}^{-N}/{}_2}^{†}P+{\hat{\mathcal{L}}}_{{}^N/{}_2}^{†}P,$$

$${\hat{\mathcal{L}}}_{a,a+1}^{†}{P}_t^{(N)}({\underline{\xi }}_N)={\int }_{-{\xi }_{a+1}}^{{\xi }_a}\text{upd}\eta K({\xi }_a-\eta ,{\xi }_{a+1}+\eta \to {\xi }_a,{\xi }_{a+1})P_t^{(N)}({\underline{\xi }}_N^{a,a+1,\eta })-\nu P_t^{(N)}({\underline{\xi }}_N),$$

$${\partial }_{t}A=\hat{\mathcal{L}}A=\sum _{a={}^{-N}/{}_2}^{{}^N/{}_2-1}{\hat{\mathcal{L}}}_{a,a+1}A+{\hat{\mathcal{L}}}_{{}^{-N}/{}_2}A+{\hat{\mathcal{L}}}_{{}^N/{}_2}A,$$

$$\begin{gathered} {\hat{\mathcal{L}}}_{a.a+1}A({\underline{\xi }}_N)={\int }_{-{\xi }_{a+1}}^{{\xi }_a}\text{upd}\eta K({\xi }_a,{\xi }_{a+1}\to {\xi }_a-\eta ,{\xi }_{a+1}+\eta ) \\ \times [A({\underline{\xi }}_N^{a,a+1,\eta })-A({\underline{\xi }}_N)]. \end{gathered}$$

$$P_t^{(N)}({\underline{\xi }}_N)=\sum _{{\underline{n}}_N}{\gamma }_{{\underline{n}}_N}\prod _{i={}^{-N}/{}_2}^{{}^N/{}_2}{p}_{{\nu }_i,{\beta }_i}({\xi }_i)J_{n_i}^{{\nu }_i}({\beta }_i{\xi }_i),$$

$$J_n^{\nu }(x)$$

$$\begin{array}{rl}x& =\nu J_0^{\nu }(x)-\sqrt{\nu }{J}_1^{\nu }(x),\\ x^2& =\nu (\nu +1)J_0^{\nu }(x)-2\sqrt{\nu }(\nu +1)J_1^{\nu }(x)\\ & +\sqrt{2\nu (\nu +1)}{J}_2^{\nu }(x).\end{array}$$

$${\gamma }_{0,\dots ,0}=1,$$

$${\gamma }_{0,\dots ,0,\underset{i}{\underbracket{1}},0,\dots ,0}=0$$

$$\frac{\text{upd}}{\text{upd}t}{\xi }_i$$

$$=\frac{1}{2\tau }\left(\frac{{\xi }_{i+1}}{{\nu }_{i+1}}-2\frac{{\xi }_i}{{\nu }_i}+\frac{{\xi }_{i-1}}{{\nu }_{i-1}}\right),$$

$$\frac{\text{upd}}{\text{upd}t}{\beta }_i=\frac{1}{2\tau {\nu }_i}({\beta }_{i+1}^{-1}-2{\beta }_i^{-1}+{\beta }_{i-1}^{-1}).$$

$$\underset{t\to \infty }{\lim }{\beta }_i^{-1}=\frac{1}{2}({\beta }_{+}^{-1}+{\beta }_{-}^{-1})+\frac{i}{N+2}({\beta }_{+}^{-1}-{\beta }_{-}^{-1}),$$

$$\underset{t\to \infty }{\lim }{⟨f_1({\xi }_i,{\xi }_{i+1})⟩}_t=-\frac{1}{2\tau }\frac{{\beta }_{+}^{-1}-{\beta }_{-}^{-1}}{N+2},$$

$$i\le j-2$$

$$\begin{gathered} \frac{\text{upd}}{\text{upd}t}{\xi }_i{\xi }_j={\xi }_i[f_1({\xi }_{j-1},{\xi }_j)-f_1({\xi }_j,{\xi }_{j+1})] \\ +{\xi }_j[f_1({\xi }_{i-1},{\xi }_i)-f_1({\xi }_i,{\xi }_{i+1})], \end{gathered}$$

$$\begin{gathered} \frac{\text{upd}}{\text{upd}t}{\xi }_i{\xi }_{i+1}={\xi }_{i+1}{f}_1({\xi }_{i-1},{\xi }_i)+({\xi }_i-{\xi }_{i+1})f_1({\xi }_i,{\xi }_{i+1}) \\ -{\xi }_i{f}_1({\xi }_{i+1},{\xi }_{i+2})-f_2({\xi }_i,{\xi }_{i+1}), \end{gathered}$$

$$\begin{gathered} \frac{\text{upd}}{\text{upd}t}{\xi }_i^2=2{\xi }_i[f_1({\xi }_{i-1},{\xi }_i)-f_1({\xi }_i,{\xi }_{i+1})]+f_2({\xi }_{i-1},{\xi }_i) \\ +f_2({\xi }_i,{\xi }_{i+1}). \end{gathered}$$

$$\begin{array}{rl}{\tilde{\gamma }}_{i:2}& =\frac{1}{\sqrt{\frac{1}{2}{\nu }_i(1+{\nu }_i)}}{\beta }_i^{-2}{\gamma }_{i:2},\\ {\tilde{\gamma }}_{i:1,j:1}& =\frac{1}{\sqrt{{\nu }_i{\nu }_j}}{\beta }_i^{-1}{\beta }_j^{-1}{\gamma }_{i:1,j:1},\end{array}$$

$$\begin{gathered} \tau {\nu }_i({\nu }_i+1)\frac{\text{upd}}{\text{upd}t}{\tilde{\gamma }}_{i:2}=-\frac{4}{3}{\beta }_i^{-1}({\beta }_{i-1}^{-1}+{\beta }_{i+1}^{-1}-{\beta }_i^{-1}) \\ +\frac{2}{3}({\beta }_{i-1}^{-2}+{\beta }_{i+1}^{-2})+\frac{2}{3}({\tilde{\gamma }}_{i-1:2}+{\tilde{\gamma }}_{i+1:2})-2({\nu }_i+\frac{1}{3}){\tilde{\gamma }}_{i:2} \\ +({\nu }_i-\frac{1}{3})({\tilde{\gamma }}_{i-1:1,i:1}+{\tilde{\gamma }}_{i:1,i+1:1}), \end{gathered}$$

$$\begin{gathered} 2\tau {\nu }_i{\nu }_{i+1}\frac{\text{upd}}{\text{upd}t}{\tilde{\gamma }}_{i:1,i+1:1}=\frac{1}{3}(2{\beta }_i^{-1}{\beta }_{i+1}^{-1}-{\beta }_i^{-2}-{\beta }_{i+1}^{-2}) \\ +{\nu }_{i+1}{\tilde{\gamma }}_{i-1:1,i+1:1}+{\nu }_i{\tilde{\gamma }}_{i:1,i+2:1}+({\nu }_i-\frac{1}{3}){\tilde{\gamma }}_{i:2} \\ +({\nu }_{i+1}-\frac{1}{3}){\tilde{\gamma }}_{i+1:2}-2({\nu }_i+{\nu }_{i+1}-\frac{1}{3}){\tilde{\gamma }}_{i:1,i+1:1}, \end{gathered}$$

$$\begin{gathered} 2\tau {\nu }_i{\nu }_j\frac{\text{upd}}{\text{upd}t}{\tilde{\gamma }}_{i:1,j:1}={\nu }_i({\tilde{\gamma }}_{i:1,j-1:1}+{\tilde{\gamma }}_{i:1,j+1:1}) \\ +{\nu }_j({\tilde{\gamma }}_{i-1:1,j:1}+{\tilde{\gamma }}_{i+1:1,j:1})-2({\nu }_i+{\nu }_j){\tilde{\gamma }}_{i:1,j:1}. \end{gathered}$$

$${\nu }_i\equiv \nu ={\epsilon }^{-1}\frac{N+3}{N+2},$$

$$i<j$$

$$\begin{gathered} \underset{t\to \infty }{\lim }{\gamma }_{i:1,j:1}\simeq {\beta }_i{\beta }_j({\beta }_{+}^{-1}-{\beta }_{-}^{-1}{)}^2\frac{N+3}{(N+2{)}^2} \\ \times \left[\left(\frac{1}{2}+\frac{i}{N+2}\right)\left(\frac{1}{2}-\frac{j}{N+2}\right)\right], \end{gathered}$$

$$\begin{gathered} \underset{t\to \infty }{\lim }{\gamma }_{i:2}\simeq \frac{{\beta }_i^2}{\sqrt{2}}({\beta }_{+}^{-1}-{\beta }_{-}^{-1}{)}^2\frac{N+3}{(N+2{)}^2} \\ \times \left[\frac{1}{4}-\frac{i^2}{(N+2{)}^2}+\frac{1}{6(N+2)}\right]. \end{gathered}$$

Full text

Real space fluctuations associated with nonequilibrium states are typically long-ranged, and thus exhibit qualitative differences with respect to their equilibrium counterparts, which, apart from the vicinity of critical points, are short-ranged. A large body of experimental evidence, in particular using small-angle light-scattering experiments, points to the generic character of these correlations, which have been paralleled by a number of theoretical developments; see reviews in [1, 2, 3, 4].

At a more fundamental level, these works have spurred interest in model lattice systems which are more easily amenable to rigorous results [5]. Long-range correlations were thus shown to arise broadly in lattice gases modeling mass transport [6]; see also [7] for a review. Nonetheless, it should be noted that the nonequilibrium steady states of specific models such as zero-range processes factorise so that correlations may indeed be absent [8, 9].

The correlations we are here more specifically concerned with are the pair correlations measured along the direction of a temperature gradient, which exhibit a simple bi-linear spatial dependence and thus encompass the entire system [10]. From a theoretical viewpoint, they emerge from the inverse Laplacian in one-dimensional lattices [6]. More generally, their presence has been inferred in the framework of multivariate stochastic models [11].

As shown by Bertini et al. [12], the Kipnis–Marchioro–Presutti (KMP) model of heat conduction [13] displays such correlations; they are positive and proportional to the square of the overall temperature gradient, but may, however, be small in that they scale uniformly with the inverse system size, consistent with the simple exclusion model [6], albeit with the opposite sign. The KMP model is a particularly simple model with local energy-conserving interactions. It consists of a one-dimensional chain of two-degrees of freedom harmonic oscillators which exchange energy among nearest neighbours through stochastic interactions at uniform rate. Under the application of a temperature gradient across the system, the nonequilibrium steady state exhibits a linear temperature gradient and sustains a heat current, given, in non-dimensional units, by minus the local temperature gradient multiplied by one half the interaction rate. The KMP model in fact lends itself to a number of rigorous results, including, in particular, the analytic characterisation of macroscopic energy fluctuations about the nonequilibrium steady state [14].

Beyond mere energy exchange models with a single conserved quantity, similar features of the correlation functions were demonstrated in a class of stochastic models coupling both mass and heat transport [15]. Another more recent example sharing similar properties of the correlation functions is a model of heat conduction with two conserved quantities [16].

In a recent publication [17], the author presented a systematic characterisation of the nonequilibrium steady state of the KMP model, within the framework of a larger class of energy exchange models having the gradient property (see, e.g., [14]), which also includes so-called Brownian energy processes [18, 19, 20]. This study yields, among other results, explicit expressions of the pair correlation functions of these models, which all exhibit long-range correlations of nearly identical shapes.

In these models, the degrees of freedom per cell, i.e. among which the cell’s energy is shared, become a continuous shape parameter, which may change with every cell. Meanwhile, a common feature is that, whenever two cells interact, all the degrees of freedom (or their continuous counterparts) are involved in the energy exchange process. As the degrees of freedom increase, however, so do the energies stored in the cells (at constant temperature). It therefore seems appropriate to take advantage of this feature and introduce a different class of energy exchange models presenting a new variation on the original KMP model, which lets the inverse of the degrees of freedom act like a small parameter, thus weakening the interactions between two energy cells.

Restricting our attention to even integer degrees of freedom per cell, we assume that the possibly many degrees of freedom are kept in a state of relative equilibrium at the cell’s energy and posit that, whenever an interaction takes place, only two among the degrees of freedom take part in the energy exchange process, carrying with them a random fraction of the cell’s energy. A remarkable consequence of this choice is that, as the degrees of freedom per cell become large, the local temperatures, given by the ratios of the local energies to the corresponding degrees of freedom, evolve deterministically in time, free of fluctuations. Simultaneously, while long-range correlations between energy cells grow, the energies involved in the interactions, whose scales are prescribed by the local temperatures, become effectively uncorrelated random variables, sampled from exponential distributions.

1 A modified Kipnis–Marchioro–Presutti model

The original KMP model [13] consists of a system of individual cells of two degrees of freedom oscillators on a one-dimensional lattice which are let to interact stochastically at uniform rate so as to redistribute their energies uniformly.

Here, we consider a generalisation of this model whereby each cell with index $i$ consists of a mechanical system of $2\nu_{i}$ identical degrees of freedom, with arbitrary integer numbers $\nu_{i}\geq 1$. Such a system could, for instance, be a collection of $\nu_{i}$ coupled two-dimensional oscillators or a two-dimensional gas of $\nu_{i}$ hard discs. These internal degrees of freedom are assumed to equilibrate on a fast, negligible timescale. We further assume a form of interaction among neighbouring cells such that, at uniform rate, two pairs of degrees of freedom (or two-dimensional “particles”) selected at random, one in each cell, redistribute their energies uniformly among themselves, subsequently equilibrating with the degrees of freedom in their respective cells. The class of models thus constructed includes the KMP model as the particular case $\nu_{i}\equiv 1$ for all $i$.

Let $\underline{\xi}_{N}\equiv\{\xi_{{}^{-N}\!\!/\!{}_{2}},\dots,\xi_{{}^{N}\!\!/\!{}_{2}}\}$, $\xi_{i}\in\mathbb{R}_{+}$, denote the collection of energy variables of a system of $N+1$ cells, with fixed associated local half degrees of freedom $\nu_{i}$. The stochastic kernel governing the interactions between cells $a$ and $b$, $|a-b|=1$, is

[TABLE]

where $\,\Theta_{\textsc{h}}\!(.)$ denotes the Heaviside step function, $\tau$ is an arbitrary time scale, and $Q_{\nu,\xi}(\epsilon)$ is the marginal energy distribution $\epsilon$ of a single pair of degrees of freedom in microcanonical equilibrium among $2\nu$ degrees of freedom with total energy $\xi$, viz. the Dirac delta distribution $\,\delta_{\textsc{d}}\!(\xi-\epsilon)$ if $\nu=1$ and, otherwise,

[TABLE]

We further note that, as follows from a simple calculation, the kernel (1) satisfies the detailed balance condition111We let $\underline{\xi}_{N}^{a,b,\eta}$ denote the collection of energy variables $\underline{\xi}_{N}$ with elements $a$ and $b$ respectively changed to $\xi_{a}-\eta$ and $\xi_{b}+\eta$.,

[TABLE]

with microcanonical equilibrium distribution, such that $\xi_{{}^{-N}\!\!/\!{}_{2}}+\dots+\xi_{{}^{N}\!\!/\!{}_{2}}=\beta^{-1}\nu_{\textsc{t}}$, where $\nu_{\textsc{t}}=\sum_{i}\nu_{i}$ is half the total number of degrees of freedom in the system, which is specified by the Dirichlet distribution,

[TABLE]

When $N\to\infty$, this distribution tends to the product of Gamma distributions with inverse temperature222The Boltzmann constant is set to unit so temperatures and energies have the same unit $\beta$,

[TABLE]

2 Kernel moments

The moments of the kernel are the functions of two energy variables $\xi_{a}$ and $\xi_{b}$, $|a-b|=1$,

[TABLE]

Of particular interest are

• (i)

the zeroth moment, $f_{0}(\xi_{a},\xi_{b})=\tau^{-1}$, which is the uniform rate of interaction among neighbouring energy cells,

• (ii)

the first moment,

[TABLE]

which is the average energy exchanged through interactions between the two cells with energies $\xi_{a}$ and $\xi_{b}$, and

• (iii)

the second moment, given by

[TABLE]

Equation (7) establishes the gradient property of the model; see Ref. [21]. In a nonequilibrium steady state, the average of the ratio $\xi/\nu$ between the cell’s energy and half the degrees of freedom it contains corresponds to a local temperature. Taking the ensemble average of equation (7) with respect to the nonequilibrium stationary distribution, we see that the stationary current corresponds to minus the local temperature gradient multiplied by one half the interaction rate, which is the heat conductivity of the model, identical to that of the KMP process.

3 Time-dependent energy distribution

Let $P_{t}^{(N)}(\underline{\xi}_{N})$ denote a time-dependent probability density on the energy configurations of $N+1$ cells. Given thermal boundary conditions at cells $\pm(N+2)/2$ with inverse temperatures $\beta_{\pm}$ and (independent) stationary energy distributions

[TABLE]

its evolution in time is prescribed by the master equation,

[TABLE]

which involves the local exchange operators $\widehat{\mathcal{L}}^{\dagger}_{a,a+1}$, {widetext}

[TABLE]

{floatequation}

see equation (11) on 11.

and their thermal boundary counterparts $\widehat{\mathcal{L}}^{\dagger}_{{}^{-N}\!\!/\!{}_{2}}$ and $\widehat{\mathcal{L}}^{\dagger}_{{}^{N}\!\!/\!{}_{2}}$, which are integrated with respect to the energy distribution of the thermal bath.

Observables $A\equiv A(\underline{\xi}_{N})$ evolve in time under the action of the adjoint operator, $\widehat{\mathcal{L}}$,

[TABLE]

where

[TABLE]

In [17], we expanded the nonequilibrium steady states of similar processes about the local equilibria (5) in terms of orthonormal polynomials and showed that such expansions enable the characterisation of correlations functions. Here we extend this scheme to time-dependent distributions $P_{t}^{(N)}$. Letting $\underline{n}_{N}$ denote the set of $N+1$ integer indices, $\underline{n}_{N}=\{n_{{}^{-N}\!\!/\!{}_{2}},\dots,n_{{}^{N}\!\!/\!{}_{2}}\}$, we thus write

[TABLE]

where the sum over the set of indices runs from [math] to $\infty$ for every index, and the polynomials $J^{\nu}_{n}$,

[TABLE]

define a complete set of orthonormal polynomials with respect to the weight function $p_{\nu,\beta=1}(x)$ (5), derived from the generalised Laguerre polynomials of parameter $\nu-1$, $L^{\nu-1}_{n}(x)$. In particular, we can write:

[TABLE]

The parameters $\beta_{i}$ and coefficients $\gamma_{\underline{n}_{N}}$ in the expansion (14) are implicit functions of time whose values are determined by considering the time-evolution of the energy moments. More specifically, letting $r$ denote the degree of a set of indices $\underline{n}_{N}$, $r=\sum_{i}n_{i}$, the set of parameters $\beta_{i}$ and coefficients $\gamma_{\underline{n}_{N}}$ associated with polynomials of degree $r$ are obtained as the solution of a set of linear differential equations, closed degree by degree.

We here restrict our attention to coefficients of degree $r\leq 2$ and begin by noting that, on the one hand, by normalisation of the stationary distribution, we must have

[TABLE]

while, on the other hand, for all $i$, the identities

[TABLE]

follow from the requirement333The notation $\left\langle\cdot\right\rangle_{t}$ stands for integration of the argument with respect to $P_{t}^{(N)}$. $\left\langle\beta_{i}\,\xi_{i}\right\rangle_{t}=\nu_{i}$. For ease of notation, we denote the above elements by $\gamma_{i:1}$, meaning that the $i$th index is unity and all the others are zero. Similarly, degree-$2$ coefficients correspond to all combinations of $\gamma_{i:2}$ and $\gamma_{i:1,j:1}$, $i<j$.

3.1 Degree-$1$ contributions

Letting $A(\underline{\xi}_{N})=\xi_{i}$, $-{}^{N}\!\!/\!{}_{2}\leq i\leq{}^{N}\!\!/\!{}_{2}$, in equation (12), we obtain the energy conservation law,

[TABLE]

where, at the boundaries, the ratios $\xi_{i\pm 1}/\nu_{i\pm 1}$ are replaced by $\beta_{\pm}$. Averaging this expression with respect to the time-dependent state (14), yields the differential equation

[TABLE]

The solution matching the thermal boundary conditions (9), $\beta_{\pm(N+2)/2}=\beta_{\pm}$, is easily found in terms of the eigenvalues and eigenfunctions of the discrete Laplace operator in one dimension; see [22, Lemma 6.1]. Its asymptotic solution in time is the linear temperature profile,

[TABLE]

with the associated stationary current,

[TABLE]

consistent with Fourier’s law and a uniform heat conductivity equal to one half the energy exchange rate.

3.2 Degree-$2$ contributions

We let $A(\underline{\xi}_{N})=\xi_{i}\,\xi_{j}$, where $-{}^{N}\!\!/\!{}_{2}\leq i\leq j\leq{}^{N}\!\!/\!{}_{2}$, and obtain, after substituting in equation (12), the three contributions:

[TABLE]

Letting

[TABLE]

we obtain, after averaging equation (24) with respect to the time-dependent state (14), the contributions, when $i=j$,

[TABLE]

Although these equations are very similar to those obtained for the KMP process (which corresponds to $\nu_{i}\equiv 1$), they now carry non-trivial dependencies on the degrees of freedom.

Focusing to start with on the stationary solutions of equations (26), we note that the special bi-linear solutions with trivial Fourier modes that are found in the KMP model for any system size $N$, provided the energy distributions of the thermal baths are modified to include second degree correlations [12, 17], cease to be valid when $\nu_{i}\neq 1$. However, for uniform and large degrees of freedom per cell, we can let

[TABLE]

and find, to leading order in $\nu^{-1}$, the approximate solutions,

[TABLE]

With this choice of parameters and as long as the energy distributions of the thermal baths are modified to account for the presence of the third term, uniform in the cell index $i$, in the square brackets on the right-hand side of (28b), the asymptotic second degree coefficients $\gamma_{i:1,j:1}$ and $\gamma_{i;2}$ have non-trivial Fourier components of order $\epsilon$.

Assuming the more conventional form (9) of the energy distributions associated with the thermal baths at the boundaries, we could resort to Fourier transforms to obtain a systematic characterisation of the second-degree coefficients. Notwithstanding its precise outcome, we observe that we must have $\gamma_{i,j}\propto(N+2)^{-1}$, which scales with the inverse system size and is independent of the degrees of freedom. By equation (16), this implies that the second degree correlations grow with $\sqrt{\nu_{i}\,\nu_{j}}$. At constant temperature, however, the energies, $\xi_{i}$, are expected to scale linearly with half the degrees of freedom $\nu_{i}$. Pair correlations thus grow slower than the products of the corresponding energies with the degrees of freedom.

We therefore expect the correlations between pairs of energy per half degrees of freedom to decrease with the inverse square root of the product of their respective degrees of freedom. Indeed, since the degrees of freedom in every cell are maintained in equilibrium throughout the process, it is easy to express the correlations between the energies $\epsilon_{i}$ of the degrees of freedom involved in the exchanges in terms of those of the cells $\xi_{i}$ and coefficients (25). Thus, if $i<j$,

[TABLE]

Time-dependent correlations must therefore vanish as the degrees of freedom increase, irrespective of the system size $N$. This is illustrated in Fig. 1 for a single-cell system and unit overall temperature gradient, with initial energy sampled from an equilibrium distribution at temperature $\tfrac{1}{2}(\beta_{+}^{-1}+\beta_{-}^{-1})$, which coincides with its temperature in the nonequilibrium steady state, and is therefore constant in time. The quantity plotted on the vertical axis is $\nu_{0}$ times the time-dependent average of the second order correlation (29c), i.e. $\nu_{0}\,\widetilde{\gamma}_{0:2}(t)$, evaluated by repeating the same measurements for up to $10^{9}$ realisations of the process. The overall timescale is divided up into 100 equal increments and contracted by $\nu_{0}$; see below.

Solving (26a), for the single-cell system at constant temperature $\beta_{0}^{-1}=\tfrac{1}{2}(\beta_{+}^{-1}+\beta_{-}^{-1})$, we simply have

[TABLE]

which, multiplied by $\nu_{0}$, corresponds to the dashed-white curves shown in Fig. 1 for different values of $\nu_{0}$.

As the numbers of degrees of freedom per cell become large, while second and higher-order correlations vanish, the distribution (2) tends to a simple exponential,

[TABLE]

Fluctuations about the local temperatures $\beta_{i}^{-1}=\lim_{\nu\to\infty}\xi_{i}/\nu$ which evolve in time according to the discrete heat equation (21) are then washed out. In this regime, however, if $\tau$ is kept constant, the timescale of temperature relaxation to its stationary profile, proportional to $\nu_{i}$ in equation (21), diverges. Assuming for simplicity $\nu_{i}\equiv\nu$ for all $i$, this issue can be remedied by letting the rate of energy exchanges scale with half the degrees of freedom, $\tau^{-1}=\nu$. The price we pay for this is that the kernel moments (6) scale with $\nu$, as does the heat conductivity. Nevertheless the ratio between the heat conductivity and energy exchange rate remains constant, equal to $\tfrac{1}{2}$.

The description of the process thus boils down to the time-evolution of its temperature profile (21). Moreover, the energy exchanges driving this time-evolution involve energy pairs drawn independently of each other from the exponential distributions (31) with the time-dependent local inverse temperatures $\beta_{i}$.

An illustration is provided in Fig. 2 where the spreads of measured values of the ratios $\xi_{i}/\nu$, $\nu=2^{15}$, are plotted as functions of time for a system of $9$ cells ($N=8$) in contact with two thermal baths at respective temperatures $\beta_{-}^{-1}=\tfrac{1}{2}$ and $\beta_{+}^{-1}=\tfrac{3}{2}$, initially starting with zero energy in every cell. The thickness of the curves is twice the standard deviation of $\xi_{i}/\nu$ with respect to $\beta_{i}^{-1}$, proportional to $1/\sqrt{\nu}$. The measured curves are compared with the time-dependent analytic solutions of (21) (dashed white curves). The heights of the horizontal black dashed lines are the asymptotic stationary values.

To better appreciate how the spread of $\xi_{i}/\nu$ about the local temperature $\beta_{i}^{-1}$ in (21) decreases as $\nu$ increases, we plot in Fig. 3 measurements of these quantities for conditions similar to that of Fig. 2, except for the values of $\nu$, which vary according to $\nu=2^{k}$, $k=0,1,\dots,15$. In particular, the $k=0$ lightblue region corresponds to the energy fluctuations of the KMP model. Since the number of events per unit time is proportional to $\nu$, the CPU integration time doubles for each increment of $\nu$. In contrast, the spread of $\xi_{i}/\nu$ decreases with $1/\sqrt{\nu}$.

4 Conclusions

The nonequilibrium states of simple stochastic models of transport generically display long-range correlations. The elementary model of heat conduction under a temperature gradient described here illustrates how the inclusion of internal degrees of freedom not directly involved in the transport process provides a simple mechanism that undercuts such correlations, even as the local temperature gradient is kept fixed. In hindsight, we believe this observation provides a new perspective on a somewhat puzzling yet successful dichotomy of nonequilibrium states, which is that of the coexistence of two seemingly antagonistic properties: the prevalence of long-range correlations contrasted with the remarkable effectiveness of the local thermodynamic equilibrium hypothesis.

Whereas the energies stored in every cell grow linearly with the local degrees of freedom, the growth of the energy pair correlations is similar and therefore slower than the growth of the product of the two energies. A straightforward consequence is that correlations between the energies attached to the two pairs of degrees of freedom directly involved in the energy exchanges decay with the square root of the product of the degrees of freedom of the two cells.

As the degrees of freedom per cell become large, the interaction rate grows faster and the definiteness of the local temperatures as a dynamical quantity—given by the ratios between the energies of the corresponding cells and degrees of freedom—sharpens up while fluctuations disappear. The energies that take part in the transfer process are random variables drawn from exponential distributions whose scales are specified by the local temperatures, i.e. local equilibrium distributions.

Acknowledgements.

The author is financially supported by the FRS-FNRS.