The Correlation Production in Thermodynamics

Abstract

Macroscopic many-body systems always exhibit irreversible behaviors. For example, the gas always tend to fill up the unoccupied area until reaching the new uniform distribution, together with the irreversible entropy increase. However, in principle, the underlying microscopic dynamics of the many-body system, either the (quantum) von Neumann or (classical) Liouville equation, guarantees the entropy of an isolated system does not change with time, which is quite confusing comparing with the macroscopic irreversibility. Notice that, due to the restrictions in practical measurements, usually it is the partial information (e.g., marginal distribution, few-body observable expectation) that is directly accessible to our observations, rather than the full ensemble state. But indeed such partial information is sufficient to give most macroscopic thermodynamic quantities, and they exhibits irreversible behaviors. At the same time, there is some correlation entropy hiding in the full ensemble, i.e., the mutual information between different marginal distributions, but difficult to be sensed in practice. We notice that such correlation production is indeed closely related to the macroscopic entropy increase in the standard thermodynamics. In open systems, the irreversible entropy production of the open system can be proved to be equivalent with the correlation production between the open system and its environment. During the free diffusion of an isolated ideal gas, the correlation between the spatial and momentum distributions is increasing monotonically, and it could well reproduce the entropy increase result in the standard thermodynamics. In the presence of particle collisions, the single-particle distribution always approaches the Maxwell-Boltzmann distribution as its steady state, and its entropy increase indeed indicates the correlation production between the particles. In all these examples, the total entropy of the whole isolated system keeps constant. In this sense, the macroscopic irreversibility and the reversible microscopic dynamics coincide with each other. arXiv:1808.00452v2 [cond-mat.stat-mech]

Entropy Production in Open Systems: The Predominant Role of Intraenvironment Correlations

“…It was often held [4][5][6][7][8][9][10][11][12] that the relative entropy D[ρ E (t)||ρ eq E ] is negligible for large thermal reservoirs. Based on this assumption, some recent papers even directly identified the entropy production with the mutual information between the system and the environment [6,10]. In this Letter we show, however, that in small open systems driven out of equilibrium the opposite is the case.…”

“…. This conclusion may be surprising because it was often held [4][5][6][7][8][9][10][11][12] that the term D[ρ E (t)||ρ eq E ] is of the second order to the change of the density matrix of the environment ∆ρ E = ρ E (t) − ρ eq E , and therefore can be neglected for large thermal reservoirs. However, whereas such order-of-magnitude arguments are valid for numbers, they should be applied with care when considering complex, multi-element structures, such as density matrices; this is because a sum of many small contributions can still be significant.…”

The hierarchy recurrences in local relaxation

“…It turns out the total correlation approximately exhibits a monotonic increasing behavior, and the increasing curve becomes more and more "smooth" with the increase of the bath size. Thus, the total correlation exhibits a quite similar behavior as the irreversible entropy increase in the standard thermodynamics [16][17][18][19]. In contrast, the whole N-body system always keeps a constant due to the unitary evolution, and the entropy of each single TLS increases and decreases from time to time.…”

“…Notice that, in practical observations, indeed the full Nbody state is usually not directly accessible for local measurements, and it is the few-body observables that can be directly measured [5,19,24]. Therefore, here we consider the dynamics of the total correlation entropy of the N-body state ρ(t), that is [12][13][14][15],…”

Entropy production and correlation spreading in the interaction between particle detector and thermal baths

“…One natural question to ask is how these two terms contribute to entropy production. It was often held that the relative entropy is negligible for large thermal reservoirs [4][5][6][7][8][9][10][11][12], compared with the MI. However, recently it was found that while entropy production could be time-extensive, the MI is strongly bounded from above by the Araki-Lieb inequality [13], so the entropy production could also predominantly come from the relative entropy that measures the displacement of the environment from equilibrium state ρ E [14].…”