Microscopic diagonal entropy and its connection to basic thermodynamic relations

Journal Article · Tue Feb 15 04:00:00 EST 2011 · Annals of Physics (New York)

DOI:https://doi.org/10.1016/j.aop.2010.08.004· OSTI ID:21579840

Polkovnikov, Anatoli ^[1]

---

1. Department of Physics, Boston University, Boston, MA 02215 (United States)

We define a diagonal entropy (d-entropy) for an arbitrary Hamiltonian system as S{sub d}=-{Sigma}{sub n{rho}nn}ln{rho}{sub nn} with the sum taken over the basis of instantaneous energy states. In equilibrium this entropy coincides with the conventional von Neumann entropy S{sub n} = -Tr{rho} ln {rho}. However, in contrast to S{sub n}, 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 S{sub d} behaves consistently with expectations from thermodynamics.

• OSTI ID: 21579840
• Journal Information: Annals of Physics (New York), Journal Name: Annals of Physics (New York) Journal Issue: 2 Vol. 326; ISSN 0003-4916; ISSN APNYA6
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
ADIABATIC PROCESSES
ENERGY SPECTRA
ENTROPY
EQUILIBRIUM
HAMILTONIANS
LOCALITY
MATHEMATICAL OPERATORS
MECHANICS
PHYSICAL PROPERTIES
QUANTUM OPERATORS
SPECTRA
STATISTICAL MECHANICS
THERMODYNAMIC PROPERTIES
THERMODYNAMICS