Surfaces of Constant Temperature in Time

TL;DR

This paper introduces a vectorized approach to characteristic times in systems, deriving a relationship between energy and time norms, and uses this to determine surfaces of constant temperature applicable to various empirical systems.

Contribution

It presents a novel vectorized framework for characteristic times, linking energy-time relationships via $L_2$ norms and deriving surfaces of constant temperature from microscopic data.

Findings

Derived a relationship between energy and time norms in vector form

Established surfaces of constant temperature in terms of time coordinates

Applicable to both simulated and real empirical systems

Abstract

The inverse relationship between energy and time is as familiar as Planck's constant. From the point of view of a system with many states, perhaps a better representation of the system is a vector of characteristic times (one per state) for example, a canonically distributed system. In the vector case the inverse relationship persists, this time as a relation between the $L_2$ norms. That relationship is derived herein. An unexpected benefit of the vectorized time viewpoint is the determination of surfaces of constant temperature in terms of the time coordinates. The results apply to all empirically accessible systems, that is situations where details of the dynamics are recorded at the microscopic level of detail. This includes all manner of simulation data of statistical mechanical systems as well as experimental data from actual systems (e.g. the internet, financial market data) where statistical physical methods have been applied.