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arXiv:1508.04864 · Foundations of Physics

Identities for Entropy Change Associated with the Time-Evolution of an Open System

Hiroki Majima, Akira Suzuki

Published Apr 28, 2015• 3 citations
arXivHTMLPDF
DOI
10.1007/s10701-015-9908-3

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Speculation on Photon Entropy during Refraction

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Entropy is usually associated with probabilities (e.g. Shannon’s entropy= S = - Sum over i Probability(i) ln{Probability(i)} which suggests there is some uncertainty in the problem, captured by S. Entropy, however, also seems to be associated with the evolution of a system in time. Historically, entropy was introduced through the first law of thermodynamics: dE = TdS - PdV, before there was any formula involving probabilities. Furthermore, a constant entropy, which applies to an adiabatic process, means the process is reversible in time. In this note, we consider the case of a photon which is represented by a quantum wavefunction exp(-iEt+ip dot r), but is also reversible in the case of reflection/refraction, even from a moving interface. The wavefunction, we argue, is associated with probability in space-time and so there should be associated entropies (as we have argued in (1)), but at the same entropy should be constant as the motion is reversible. In (1), we considered Shannon’s entropy for t and x and used A1 |cos(Et)| and A2 |cos(px)| over half a cycle as probabilities. We introduce a third entropy associated with time, but not E in this note. Here, we focus on the case of refraction. For a stationary interface, E is conserved, but p changes. Thus there is a change in the uncertainty in x, but not in t and the process is reversible, with time S being constant.( The first law suggests that S=constant, which is keeping with time entropy, but not with spatial (x) entropy.) We suggest that there may be multiple types of entropy associated with different observables. We next consider the case of reflection/refraction from a moving mirror/interface. In such a case, neither E nor p is conserved for the photon. Both time and spatial (x) entropy change, but the process is reversible and entropy should be constant. We suggest there exists an entropy which is constant, but is based on: -Et+ p dot r = t |p| (-E/|p| + unit vector dot r/t) where r/t vector is minus the velocity of the moving interface and the unit vector points in the direction of momentum p. We consider an associated entropy and the first law of thermodynamics.

•Sep 29, 2023•0 citations

Time evolution of open systems

W. Peier

Physica•Mar 1, 1972•20 citations
Weak invariants are time-dependent observables with conserved expectation values. Their fluctuations, however, do not remain constant in time. On the assumption that time evolution of the state of an open quantum system is given in terms of a completely positive map, the fluctuations monotonically grow even if the map is not unital, in contrast to the fact that monotonic increases of both the von Neumann entropy and Rényi entropy require the map to be unital. In this way, the weak invariants describe temporal asymmetry in a manner different from the entropies. A formula is presented for time evolution of the covariance matrix associated with the weak invariants in cases where the system density matrix obeys the Gorini-Kossakowski-Lindblad-Sudarshan equation.
The von Neumann entropy rate for open quantum systems is, in general, written in terms of entropy production and entropy flow rates, encompassing the second law of thermodynamics. For memoryless open quantum systems, whose evolutions correspond to one-parameter quantum dynamical semigroups, we find a decomposition of the infinitesimal generator of the dynamics that unveils the fluctuation-dissipation structure of these evolutions. Thus, the fluctuation part of this decomposition allows to relate the von Neumann entropy rate with the divergence-based quantum Fisher information, at any time. Applied to continuous-in-time quantum Gaussian channels, our decomposition leads to the quantum analog of the generalized classical de Bruijn identity, thus expressing the quantum fluctuation-dissipation relation in that kind of channels. Finally, from this perspective, we analyze how stationarity arises.
Entropy is usually associated with probabilities (e.g. Shannon’s entropy= S = - Sum over i Probability(i) ln{Probability(i)} which suggests there is some uncertainty in the problem, captured by S. Entropy, however, also seems to be associated with the evolution of a system in time. Historically, entropy was introduced through the first law of thermodynamics: dE = TdS - PdV, before there was any formula involving probabilities. Furthermore, a constant entropy, which applies to an adiabatic process, means the process is reversible in time. In this note, we consider the case of a photon which is represented by a quantum wavefunction exp(-iEt+ip dot r), but is also reversible in the case of reflection/refraction, even from a moving interface. The wavefunction, we argue, is associated with probability in space-time and so there should be associated entropies (as we have argued in (1)), but at the same entropy should be constant as the motion is reversible. In (1), we considered Shannon’s entropy for t and x and used A1 |cos(Et)| and A2 |cos(px)| over half a cycle as probabilities. We introduce a third entropy associated with time, but not E in this note. Here, we focus on the case of refraction. For a stationary interface, E is conserved, but p changes. Thus there is a change in the uncertainty in x, but not in t and the process is reversible, with time S being constant.( The first law suggests that S=constant, which is keeping with time entropy, but not with spatial (x) entropy.) We suggest that there may be multiple types of entropy associated with different observables. We next consider the case of reflection/refraction from a moving mirror/interface. In such a case, neither E nor p is conserved for the photon. Both time and spatial (x) entropy change, but the process is reversible and entropy should be constant. We suggest there exists an entropy which is constant, but is based on: -Et+ p dot r = t |p| (-E/|p| + unit vector dot r/t) where r/t vector is minus the velocity of the moving interface and the unit vector points in the direction of momentum p. We consider an associated entropy and the first law of thermodynamics.