[Paper Review] Maximum entropy principle and the form of source in non-equilibrium statistical operator method

V. V. Ryazanov|ArXiv.org|Oct 23, 2009

Advanced Thermodynamics and Statistical Mechanics3 references3 citations

TL;DR

This paper applies the maximum entropy principle to determine the form of the lifetime distribution function in the non-equilibrium statistical operator (NESOM) method, replacing Zubarev's exponential distribution with more general forms. It derives distributions—including power-law and log-normal—by optimizing entropy under physical constraints, showing that the choice of lifetime distribution significantly affects non-equilibrium dynamics even in the thermodynamic limit.

ABSTRACT

It is supposed that the exponential multiplier in the method of the non-equilibrium statistical operator (Zubarev`s approach) can be considered as a distribution density of the past lifetime of the system, and can be replaced by an arbitrary distribution function. To specify this distribution the method of maximum entropy principle as in [Schönfeldt J-H, Jiminez N, Plastino A R, Plastino A, Casas M 2007 extit{Physica A} extbf{374} 573] is used. The obtained distribution is close to exponential one. Another approach to the maximum entropy principle, as in [Van der Straeten E and Beck C 2008 Phys. Rev. E extbf{78} 051101], except exponential distributions yields power-like, log-normal distributions, as well as distributions of other kind and transitions between them.

Motivation & Objective

• To generalize the non-equilibrium statistical operator (NESOM) method by replacing Zubarev's exponential lifetime distribution with arbitrary distributions derived via maximum entropy.
• To investigate how different forms of the lifetime distribution function affect the non-equilibrium behavior of systems, especially in the thermodynamic limit.
• To provide a physically motivated derivation of the weight function $ p_q(u) $ in NESOM using information-theoretic principles.
• To explore transitions between different distribution types (e.g., power-law to exponential) by modifying the entropy constraints.

Proposed method

• Uses the maximum entropy principle to optimize the entropy functional $ S = \int p_q(u) \ln p_q(u) \, du $ under physical constraints.
• Imposes constraints on the average values of observables $ \langle A_m \rangle $, energy, and a general function $ g(u) $, leading to a parametric form for $ p_q(u; \lambda_i) $.
• Derives the general form $ p_q(u; \lambda_i) = \frac{Z(t-u)^{-\lambda_1/V}}{Z(\lambda_i)} \exp\left(-\lambda_2 \frac{\sum_m F_m(t-u)\langle A_m \rangle}{V} - \lambda_3 g(u) \right) $, where $ g(u) $ encodes system-specific history dependence.
• Adjusts $ g(u) $ to generate different distribution types: $ g(u) = u $ yields exponential, $ g(u) = \ln u $ yields power-law, and $ g(u) = (\ln u)^2 $ yields log-normal distributions.
• Performs a thermodynamic limit $ N \to \infty, V \to \infty, N/V = \text{const} $, while allowing $ \varepsilon \to 0 $, to analyze the stability and physical relevance of the derived distributions.

Experimental results

Research questions

• RQ1 Can the exponential lifetime distribution in NESOM be generalized beyond the Zubarev form using maximum entropy principles?
• RQ2 How do different functional forms of the lifetime distribution $ p_q(u) $ affect the non-equilibrium dynamics of a system?
• RQ3 What physical constraints lead to power-law or log-normal lifetime distributions in the NESOM framework?
• RQ4 Is the choice of the lifetime distribution function significant in the thermodynamic limit, or does it vanish asymptotically?
• RQ5 Can transitions between different distribution types (e.g., from power-law to exponential) be modeled by adjusting the entropy constraints?

Key findings

• The maximum entropy principle yields a distribution close to exponential but not identical to Zubarev’s $ \varepsilon e^{-\varepsilon u} $, indicating a non-trivial deviation from the standard NESOM form.
• By setting $ g(u) = \ln u $, the method produces a power-law distribution for the lifetime, while $ g(u) = (\ln u)^2 $ leads to a log-normal distribution.
• The derived form $ p_q(u; \lambda_i) $ allows for a continuous family of distributions depending on the choice of $ g(u) $, enabling modeling of complex temporal behavior.
• Even in the thermodynamic limit, the form of $ p_q(u) $ influences the non-equilibrium evolution, showing that the choice of distribution is physically consequential.
• The model supports experimental observations of non-exponential lifetime distributions in systems undergoing transitions to chaos or turbulence.
• The interpretation of $ t_0 $ as a fluctuating initial time allows for slow changes in $ g(u) $, such as $ \ln(t - t_0) $, which can model evolving system-environment interactions over time.