Micro-reversibility and thermalization with collisional baths

Abstract

Micro-reversibility plays a central role in thermodynamics and statistical mechanics. It is used to prove that systems in contact with a thermal bath relax to canonical ensembles. However, a problem arises when trying to reproduce this proof for classical and quantum collisional baths, i.e. particles at equilibrium interacting with a localized system via collisions. In particular, micro-reversibility appears to be broken and some models do not thermalize when interacting with Maxwellian particles. We clarify these issues by showing that micro-reversibility needs the invariance of evolution equations under time reversal plus the conservation of phase space volume in classical and semiclassical scenarios. Consequently, all canonical variables must be considered to ensure thermalization. This includes the position of the incident particles which maps their Maxwellian distribution to the effusion distribution. Finally, we show an example of seemingly plausible collision rules that do not conserve phase-space volume, and consequently violate the second law.The first issue that we address concerns the very formulation of the micro-reversibility condition. The generic statement of micro-reversibility is that the probability to observe a transition Γ → Γ is equal to the probability of the reverse transitionΓ →Γ. Here, Γ and Γ are arbitrary microscopic states of a physical system andΓ the time-reversal state of Γ. The mathematical expression of this statement reads (we provide a more detailed description later on, in Sec. 2):However, this equality immediately poses a problem if the variable Γ is continuous. In that case, ρ(Γ |Γ) is a density in Γ , whereas ρ(Γ|Γ ) is a density inΓ. Consequently, when the two conditional probabilities are compared, one has to take into account the transformation of the volume elements dΓ and dΓ . In classical systems, Liouville's theorem warrants the conservation of volume, implying dΓ = dΓ , and resolves the problem. But micro-reversibility is also relevant for quantum and semi-classical systems with states parametrized by continuous variables Γ, such as Wigner distributions in phase space or wave packets centered around a given position and velocity [9,10]. In the first case, for instance, it has been shown that there is no equivalent Liouville-like theorem, that is, unitary evolution does not necessarily conserve the phase-space volume [11]. Therefore, it is necessary to explicitly check the micro-reversibility condition, Eq. (1), in those quantum and semi-classical scenarios.The second issue concerns the role of micro-reversibility in the relaxation of a system towards equilibrium. For a classical system in contact with a thermal bath, the micro-reversibility condition applies to micro-states Γ = (x, y) of the global system, consisting of the system itself (x) and the bath (y). The bath variables can be eliminated by multiplying Eq. (1) by the equilibrium distribution ρ eq (y) and integrating over y and y . This procedure, which we describe in detail in Sec. 2.1, is especially relevant...

Quantum collisional thermostats

“…In a series of papers [18,19], we adopted a different strategy and managed to build a repeated interaction scheme with zero work by considering a fully autonomous scenario, where the units escape from the reservoir with a random velocity given by the effusion distribution, move in space as quantum wave packets, and collide with the system without the need of an external agent. In this case, the energy to switch on and off the interaction is provided by the spatial degree of freedom of the unit.…”

Stochastic Collisional Quantum Thermometry

“…While employing WTDs in this sense clearly brings collision models closer to modelling real physical systems [ 43 , 44 , 45 ], here, we explore how introducing such randomness affects the performance of the collisional thermometry scheme. As we shall demonstrate, stochastic collision models allow us to achieve a greater range of parameter estimation over deterministic collision models without significantly sacrificing the maximal achievable precision.…”

Collision Models Can Efficiently Simulate Any Multipartite Markovian Quantum Dynamics

“…This framework, whose origins can be traced back to some important works of the previous century [1][2][3], has given birth to a pletora of "collision" or "repeated interactions" models [4][5][6][7][8][9][10][11][12][13], which have been receiving an increasing attention in recent years, especially due to their fundamental importance in the fields of quantum thermodynamics and open quantum systems. For instance, collision models have been proven useful to investigate flux rectification [14], Landauer's principle [15,16], the emergence of thermalization or non-equilibrium steady states [17][18][19][20][21][22][23][24][25][26], quantum thermometry [27], quantum batteries [28] and quantum thermal machines [29][30][31][32][33], as well as to analyze the thermodynamics of non-thermal baths [34][35][36] or in the presence of strong coupling [37]. Applications outside the field of thermodynamics include the study of open quantum optical systems [38][39][40][41], simulation of non-Markovian effects [9,…”