Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1811.11321

Generalized Boltzmann distributions for systems strongly coupled to large finite bath -- a microcanonical approach

Yu-Chen Cheng, Wenning Wang, Zhiyue Lu, Hong Qian

Published Nov 28, 2018• 0 citations
arXivHTMLPDF

Abstract

The theory of probability shows that, as the fraction $X_n/Y\to 0$, the conditional probability for $X_n$, given $X_n+Y \in h_δ:=[h, h+δ]$, has a limit law $f_{X_n}(x)e^{-ψ_n(h_δ)x}$, where $ψ_n(h_δ) $ equals to $[\partial \ln P(Y \in y_δ)/\partial y]_{y=h}$ plus an additional term, contributed from the correlation between $X_n$ and bath $Y$. By applying this limit law to an isolated composite system consisting of two strongly coupled parts, a system of interest and a large but finite bath, we derive the generalized Boltzmann distribution law for the system of interest in the exponential form of a redefined Hamiltonian and corrected Boltzmann temperature that reflects the modification due to strong system-bath coupling and the large but finite bath.

DOI
10.48550/arxiv.1811.11321

Related papers

Typicality for Generalized Microcanonical Ensembles

Peter Reimann

For a macroscopic, isolated quantum system in an unknown pure state, the expectation value of any given observable is shown to hardly deviate from the ensemble average with extremely high probability under generic equilibrium and nonequilibrium conditions. Special care is devoted to the uncontrollable microscopic details of the system state. For a subsystem weakly coupled to a large heat bath, the canonical ensemble is recovered under much more general and realistic assumptions than those implicit in the usual microcanonical description of the composite system at equilibrium.

arXiv:0710.4214•Physical Review Letters•Oct 19, 2007•268 citations

Axioms for the Boltzmann Distribution

Adam M. Brandenburger, Kai Steverson

A fundamental postulate of statistical mechanics is that all microstates in an isolated system are equally probable. This postulate, which goes back to Boltzmann, has often been criticized for not having a clear physical foundation. In this note, we provide a derivation of the canonical (Boltzmann) distribution that avoids this postulate. In its place, we impose two axioms with physical interpretations. The first axiom (thermal equilibrium) ensures that, as our system of interest comes into contact with different heat baths, the ranking of states of the system by probability is unchanged. Physically, this axiom is a statement that in thermal equilibrium, population inversions do not arise. The second axiom (energy exchange) requires that, for any heat bath and any probability distribution on states, there is a universe consisting of a system and heat bath that can achieve this distribution. Physically, this axiom is a statement that energy flows between system and heat bath are unrestricted. We show that our two axioms identify the Boltzmann distribution.

Foundations of Physics•May 1, 2019•5 citations

A REMARK ON BOLTZMANN H THEOREM

CHANG TSUNG-SDEI, 中国科学院数学研究所,北京师范大学物理系

It is pointed out that the quantum-mechanical Boltzmann H theorem in the restricted sense (i.e. with explicit consideration of the statistics of the particles constituting the system and their interactions) does not lead directly to the distribution laws for the particles among their various states as is commonly believed, but to the equality of the probability for all states belonging to the system as a whole if the system is isolated, (or to the Boltzmann factor for the probabilities of states belonging to a system at constant temperature), in agreement with the result of the generalized quantum-mechanical H theorem as given by Pauli. The distribution laws for the particles among their various states may of course be deduced from the latter.In the course of proof, we have also introduced the following minor improvements:(i) To avoid conservation of energy in microscopic processes which lead to changes in the states of the particles and thus to equilibrium, we let our system be in interaction with a heat reservoir, so that energy may be supplied by or given to it. This is tauto-mount to changing the study of the entropy for an isolated system to that of the free energy for a system at constant temperature.(ii) Interaction energies between the particles which affect the total energy of the system and thus the distribution laws for the particles among their various states are included.

Acta Physica Sinica•Jan 1, 1954•2 citations

Generalized Hamiltonian of mean-force approach to open quantum systems coupled to finite baths in thermoequilibrium

Xuerui Du, Jianhui Wang, Yongli Ma

The prevailing method for addressing strong-coupling thermodynamics typically involves open quantum systems coupled to infinite baths in equilibrium through the Hamiltonian of mean force (HMF). However, its applicability to finite baths remains limited. In this work, we transcend this limit by considering the impacts of system-bath coupling which is not only on the system but also on the finite bath feedback. When the bath and system sizes are comparable, they act as each other's effective 'bath'. We introduce a parameter α within [0,1], where α and 1-α act as weighting factors to distribute the system-bath coupling between the system and the bath. Our approach innovatively incorporates the effect of coupling on the respective effective 'bath' into both the system and bath in the statistical factor form. We generalize the HMF and propose quantum Hamiltonians of mean forces to handle the distributed coupling of the system to finite baths. The additional parameter α is determined by minimizing the joint free energy density and is a new thermodynamic quantity representing the finite bath's influence. Examining the damped quantum harmonic oscillator with a finite bath, we find that α affects both states, leading to a complex phase diagram with unique phenomena at critical temperatures T_{L} and T_{R}, including valleys, peaks, negative values, and discontinuities in entropy and specific heat. Notably, these anomalies disappear when α→1, both at high temperatures and in the thermodynamic limit, indicating the negligible influence of coupling on the bath states and reverting to the HMF. Thus, studying finite-sized baths holds substantial significance in small quantum systems.

Physical review. E•Feb 24, 2025•1 citation

From Quantum Dynamics to the Canonical Distribution: General Picture and a Rigorous Example

Hal Tasaki

Derivation of the canonical (or Boltzmann) distribution based only on quantum dynamics is discussed. Consider a closed system which consists of a mutually interacting subsystem and a heat bath, and assume that the whole system is initially in a pure state (which can be far from equilibrium) with small energy fluctuation. Under the ``hypothesis of equal weights for eigenstates,'' we derive the canonical distribution in the sense that, at sufficiently large and typical time, the (instantaneous) quantum mechanical expectation value of an arbitrary operator of the subsystem is almost equal to the desired canonical expectation value. We present a class of examples in which the above derivation can be rigorously established without any unproven hypotheses.

arXiv:cond-mat/9707253•Physical Review Letters•Feb 16, 1998•426 citations

On Work and Heat in Time-Dependent Strong Coupling

Erik M. Aurell

This paper revisits the classical problem of representing a thermal bath interacting with a system as a large collection of harmonic oscillators initially in thermal equilibrium. As is well known, the system then obeys an equation, which in the bulk and in the suitable limit tends to the Kramers–Langevin equation of physical kinetics. I consider time-dependent system-bath coupling and show that this leads to an additional harmonic force acting on the system. When the coupling is switched on and switched off rapidly, the force has delta-function support at the initial and final time. I further show that the work and heat functionals as recently defined in stochastic thermodynamics at strong coupling contain additional terms depending on the time derivative of the system-bath coupling. I discuss these terms and show that while they can be very large if the system-bath coupling changes quickly, they only give a finite contribution to the work that enters in Jarzynski’s equality. I also discuss that these corrections to standard work and heat functionals provide an explanation for non-standard terms in the change of the von Neumann entropy of a quantum bath interacting with a quantum system found in an earlier contribution (Aurell and Eichhorn, 2015).

arXiv:1705.07811•Entropy•Nov 7, 2017•23 citations
The theory of probability shows that, as the fraction Xn/Y→0X_n/Y\to 0Xn​/Y→0, the conditional probability for XnX_nXn​, given Xn+Y∈hδ:=[h,h+δ]X_n+Y \in h_δ:=[h, h+δ]Xn​+Y∈hδ​:=[h,h+δ], has a limit law fXn(x)e−ψn(hδ)xf_{X_n}(x)e^{-ψ_n(h_δ)x}fXn​​(x)e−ψn​(hδ​)x, where ψn(hδ)ψ_n(h_δ) ψn​(hδ​) equals to [∂ln⁡P(Y∈yδ)/∂y]y=h[\partial \ln P(Y \in y_δ)/\partial y]_{y=h}[∂lnP(Y∈yδ​)/∂y]y=h​ plus an additional term, contributed from the correlation between XnX_nXn​ and bath YYY. By applying this limit law to an isolated composite system consisting of two strongly coupled parts, a system of interest and a large but finite bath, we derive the generalized Boltzmann distribution law for the system of interest in the exponential form of a redefined Hamiltonian and corrected Boltzmann temperature that reflects the modification due to strong system-bath coupling and the large but finite bath.
For a macroscopic, isolated quantum system in an unknown pure state, the expectation value of any given observable is shown to hardly deviate from the ensemble average with extremely high probability under generic equilibrium and nonequilibrium conditions. Special care is devoted to the uncontrollable microscopic details of the system state. For a subsystem weakly coupled to a large heat bath, the canonical ensemble is recovered under much more general and realistic assumptions than those implicit in the usual microcanonical description of the composite system at equilibrium.
A fundamental postulate of statistical mechanics is that all microstates in an isolated system are equally probable. This postulate, which goes back to Boltzmann, has often been criticized for not having a clear physical foundation. In this note, we provide a derivation of the canonical (Boltzmann) distribution that avoids this postulate. In its place, we impose two axioms with physical interpretations. The first axiom (thermal equilibrium) ensures that, as our system of interest comes into contact with different heat baths, the ranking of states of the system by probability is unchanged. Physically, this axiom is a statement that in thermal equilibrium, population inversions do not arise. The second axiom (energy exchange) requires that, for any heat bath and any probability distribution on states, there is a universe consisting of a system and heat bath that can achieve this distribution. Physically, this axiom is a statement that energy flows between system and heat bath are unrestricted. We show that our two axioms identify the Boltzmann distribution.
It is pointed out that the quantum-mechanical Boltzmann H theorem in the restricted sense (i.e. with explicit consideration of the statistics of the particles constituting the system and their interactions) does not lead directly to the distribution laws for the particles among their various states as is commonly believed, but to the equality of the probability for all states belonging to the system as a whole if the system is isolated, (or to the Boltzmann factor for the probabilities of states belonging to a system at constant temperature), in agreement with the result of the generalized quantum-mechanical H theorem as given by Pauli. The distribution laws for the particles among their various states may of course be deduced from the latter.In the course of proof, we have also introduced the following minor improvements:(i) To avoid conservation of energy in microscopic processes which lead to changes in the states of the particles and thus to equilibrium, we let our system be in interaction with a heat reservoir, so that energy may be supplied by or given to it. This is tauto-mount to changing the study of the entropy for an isolated system to that of the free energy for a system at constant temperature.(ii) Interaction energies between the particles which affect the total energy of the system and thus the distribution laws for the particles among their various states are included.
The prevailing method for addressing strong-coupling thermodynamics typically involves open quantum systems coupled to infinite baths in equilibrium through the Hamiltonian of mean force (HMF). However, its applicability to finite baths remains limited. In this work, we transcend this limit by considering the impacts of system-bath coupling which is not only on the system but also on the finite bath feedback. When the bath and system sizes are comparable, they act as each other's effective 'bath'. We introduce a parameter α within [0,1], where α and 1-α act as weighting factors to distribute the system-bath coupling between the system and the bath. Our approach innovatively incorporates the effect of coupling on the respective effective 'bath' into both the system and bath in the statistical factor form. We generalize the HMF and propose quantum Hamiltonians of mean forces to handle the distributed coupling of the system to finite baths. The additional parameter α is determined by minimizing the joint free energy density and is a new thermodynamic quantity representing the finite bath's influence. Examining the damped quantum harmonic oscillator with a finite bath, we find that α affects both states, leading to a complex phase diagram with unique phenomena at critical temperatures T_{L} and T_{R}, including valleys, peaks, negative values, and discontinuities in entropy and specific heat. Notably, these anomalies disappear when α→1, both at high temperatures and in the thermodynamic limit, indicating the negligible influence of coupling on the bath states and reverting to the HMF. Thus, studying finite-sized baths holds substantial significance in small quantum systems.
Derivation of the canonical (or Boltzmann) distribution based only on quantum dynamics is discussed. Consider a closed system which consists of a mutually interacting subsystem and a heat bath, and assume that the whole system is initially in a pure state (which can be far from equilibrium) with small energy fluctuation. Under the ``hypothesis of equal weights for eigenstates,'' we derive the canonical distribution in the sense that, at sufficiently large and typical time, the (instantaneous) quantum mechanical expectation value of an arbitrary operator of the subsystem is almost equal to the desired canonical expectation value. We present a class of examples in which the above derivation can be rigorously established without any unproven hypotheses.
This paper revisits the classical problem of representing a thermal bath interacting with a system as a large collection of harmonic oscillators initially in thermal equilibrium. As is well known, the system then obeys an equation, which in the bulk and in the suitable limit tends to the Kramers–Langevin equation of physical kinetics. I consider time-dependent system-bath coupling and show that this leads to an additional harmonic force acting on the system. When the coupling is switched on and switched off rapidly, the force has delta-function support at the initial and final time. I further show that the work and heat functionals as recently defined in stochastic thermodynamics at strong coupling contain additional terms depending on the time derivative of the system-bath coupling. I discuss these terms and show that while they can be very large if the system-bath coupling changes quickly, they only give a finite contribution to the work that enters in Jarzynski’s equality. I also discuss that these corrections to standard work and heat functionals provide an explanation for non-standard terms in the change of the von Neumann entropy of a quantum bath interacting with a quantum system found in an earlier contribution (Aurell and Eichhorn, 2015).