Microscopic diagonal entropy and its connection to basic thermodynamic relations
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Citation (published version)Anatoli Polkovnikov. 2011. "Microscopic diagonal entropy and its connection to basic thermodynamic relations." ANNALS OF PHYSICS, Volume 326, Issue 2, pp. 486 - 499 (14). https://doi.org/10.1016/j.aop.2010.08.004
We define a diagonal entropy (d-entropy) for an arbitrary Hamiltonian system as Sd = − 𝛴𝑛 𝘗𝑛𝑛 l𝑛 𝘗𝘯𝘯 with the sum taken over the basis of instantaneous energy states. In equilibrium this entropy coincides with the conventional von Neumann entropy Sn = −Trρ ln ρ. However, in contrast to Sn, the d-entropy is not conserved in time in closed Hamiltonian systems. If the system is initially in stationary state then in accord with the second law of thermodynamics the d-entropy can only increase or stay the same. We also show that the d-entropy can be expressed through the energy distribution function and thus it is measurable, at least in principle. Under very generic assumptions of the locality of the Hamiltonian and non-integrability the d-entropy becomes a unique function of the average energy in large systems and automatically satisfies the fundamental thermodynamic relation. This relation reduces to the first law of thermodynamics for quasi-static processes. The d-entropy is also automatically conserved for adiabatic processes. We illustrate our results with explicit examples and show that Sd behaves consistently with expectations from thermodynamics.
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