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arXiv:2311.11767

Remarks on Statistical mechanics of a moving system

Jinwu Ye

Published Nov 20, 2023• 0 citations
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Abstract

In the realm of statistical mechanics, it has been established that there is no distinction between the micro-canonical and canonical ensembles in the thermodynamic limit. However, this paradigm may alter when addressing statistical mechanics in the context of a moving sample with a velocity $ v $. Our investigation reveals significant disparities between the two ensembles when considering relativistic effects up to the order of $ (v/c)^2 $. While the temperature remains the same in the former, it experiences an increase in the latter. If the system undergoes a finite-temperature phase transition, the critical temperature decreases in the co-moving frame of the latter ensemble. The implications of these findings on the thermodynamic zeroth to the third laws and the eigenstate thermalization hypothesis are analysed. The potential for the experimental detection of these novel effects in condensed matter systems are discussed.

DOI
10.48550/arxiv.2311.11767

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In the realm of statistical mechanics, it has been established that there is no distinction between the micro-canonical and canonical ensembles in the thermodynamic limit. However, this paradigm may alter when addressing statistical mechanics in the context of a moving sample with a velocity vvv. Our investigation reveals significant disparities between the two ensembles when considering relativistic effects up to the order of (v/c)2(v/c)^2(v/c)2. While the temperature remains the same in the former, it experiences an increase in the latter. If the system undergoes a finite-temperature phase transition, the critical temperature decreases in the co-moving frame of the latter ensemble. The implications of these findings on the thermodynamic zeroth to the third laws and the eigenstate thermalization hypothesis are analysed. The potential for the experimental detection of these novel effects in condensed matter systems are discussed.
The infinite-volume limit of thermodynamic functions calculated in the quantum microcanonical ensemble is shown to exist for a fairly wide class of spin systems and quantum gases. The entropy is set equal to the logarithm of the number of eigenstates in an energy interval which increases linearly with the size of the system, but is otherwise arbitrary. The limiting entropy per unit volume agrees with that calculated in the canonical formalism, and possesses certain convexity properties required for thermodynamic stability. A precise criterion, in terms of the energy spectra of large systems, is given for determining the limit of the thermodynamic entropy as the temperature approaches zero. This is not determined by the degeneracy of the ground state, contrary to the discussion of the ``third law of thermodynamics'' found in some textbooks.
We investigate eigenstate thermalization from the point of view of vanishing particle and heat currents between a few-body fermionic Hamiltonian prepared in one of its eigenstates and an external, weakly coupled Fermi-Dirac gas. The latter acts as a thermometric probe, with its temperature and chemical potential set so that there is neither particle nor heat current between the two subsystems. We argue that the probe temperature can be attributed to the few-fermion eigenstate in the sense that (i) it varies smoothly with energy from eigenstate to eigenstate, (ii) it is equal to the temperature obtained from a thermodynamic relation in a wide energy range, (iii) it is independent of details of the coupling between the two systems in a finite parameter range, (iv) it satisfies the transitivity condition underlying the zeroth law of thermodynamics, and (v) it is consistent with Carnot's theorem. For the spinless fermion model considered here, these conditions are essentially independent of the interaction strength. When the latter is weak, however, orbital occupancies in the few-fermion system differ from the Fermi-Dirac distribution so that partial currents from or to the probe will eventually change its state. We find that (vi) above a certain critical interaction strength, orbital occupancies become close to the Fermi-Dirac distribution, leading to a true equilibrium between the few-fermion system and the probe. In that case, the coupling between the Fermi-Dirac gas and few-fermion system does not modify the state of the latter, which justifies our approach a posteriori. From these results, we conjecture that for few-body systems with sufficiently strong interaction, the eigenstate thermalization hypothesis is complemented by ensemble equivalence: individual many-body eigenstates define a microcanonical ensemble that is equivalent to a canonical ensemble with grand canonical orbital occupancies.
Classical statistical mechanics tries to derive the macroscopic properties of matter starting from the mechanical laws that rule the behaviour of single particles. To describe equilibrium states, only observables accounting for correlations among states are considered, and only systems consisting of a large number of particles are taken into account. The observables are described by continuous functions on phase space, and the states are represented by linear assignments of a number to each observable. Within the framework of the canonical ensemble, one deals with mechanical systems in thermal equilibrium with a thermal reservoir, and the equilibrium state is reached as a result of the interaction with the external world. The external world may be really external, or equally well, the unrecognized internal degrees of freedom, like in the calculation of viscosity or thermal conductivity. If really only the external world is the cause we may expect some surface dependence while an internal unrecognized degree of freedom would have a volume effect, unless the interactions are long range. The microcanonical ensemble is instead introduced to study isolated mechanical systems, and the equilibrium is viewed as a temporal average, rather than as a limit. Attention is then focused on partition functions, the theorem of equipartition of energy and an elementary theory of specific heats. In the second part, the Planck derivation of the law of black-body radiation is analysed, presenting in chronological order the Kirchhoff theorem, the Stefan law, the Wien displacement law, the Rayleigh–Jeans formula and the Planck hypothesis. Further topics discussed are the Einstein and Debye quantum models for specific heats of solids. These topics prepare the ground for the introduction of quantum statistical mechanics. The third part is, in fact, devoted to the analysis of identical particles in quantum mechanics.
It is common knowledge that the microcanonical, canonical, and grand canonical ensembles are equivalent in thermodynamically large systems. Here, we study finite-size effects in the latter two ensembles. We show that contrary to naive expectations, finite-size errors are exponentially small in grand canonical ensemble calculations for translationally invariant systems in phases without long range order at finite temperature. Open boundary conditions and canonical ensemble calculations suffer from finite-size errors that scale as the inverse length scale of the system. We further show that finite-size effects are generally smallest in numerical linked cluster expansions. Our conclusions are supported by analytical and numerical analyses of classical and quantum systems.
The canonical ensemble plays a crucial role in statistical mechanics in and out of equilibrium. For example, the standard derivation of the fluctuation theorem relies on the assumption that the initial state of the heat bath is the canonical ensemble. On the other hand, the recent progress in the foundation of statistical mechanics has revealed that a thermal equilibrium state is not necessarily described by the canonical ensemble but can be a quantum pure state or even a single energy eigenstate, as formulated by the eigenstate thermalization hypothesis (ETH). Then a question raised is how these two pictures, the canonical ensemble and a single energy eigenstate as a thermal equilibrium state, are compatible in the fluctuation theorem. In this paper, we theoretically and numerically show that the fluctuation theorem holds in both of the long- and short-time regimes, even when the initial state of the bath is a single energy eigenstate of a many-body system. Our proof of the fluctuation theorem in the long-time regime is based on the ETH, while it was previously shown in the short-time regime on the basis of the Lieb-Robinson bound and the ETH [Phys. Rev. Lett. 119, 100601 (2017)0031-900710.1103/PhysRevLett.119.100601]. The proofs for these time regimes are theoretically independent and complementary, implying the fluctuation theorem in the entire time domain. We also perform a systematic numerical simulation of hard-core bosons by exact diagonalization and verify the fluctuation theorem in both of the time regimes by focusing on the finite-size scaling. Our results contribute to the understanding of the mechanism that the fluctuation theorem emerges from unitary dynamics of quantum many-body systems and can be tested by experiments with, e.g., ultracold atoms.
We have developed a theoretical formalism to introduce temperature as a parameter into the framework of non-relativistic quantum mechanics using the laws of classical thermodynamics and the canonical ensemble scheme of statistical mechanics. A self-consistent Hamiltonian has then been constructed for a given quantum many-body system which includes the effect of temperature in the form of correction terms added to the corresponding zero-temperature Hamiltonian of the system. Investigating some quantum mechanical systems with exact zero-temperature solutions including the particle-in-a-box model, the free particle, and the harmonic oscillator within our finite-temperature approach up to the first order of self-consistency has led to temperature-dependent Hamiltonians describing these systems above absolute zero without encountering any physically unacceptable brand of behavior for their wave functions and energy spectra. Results firmly support the view that a quantum mechanical system at a finite temperature behaves as if it is in a zero-temperature excited state.