Microscopic diagonal entropy and its connection to basic thermodynamic relations

Microscopic expression for the heat in the adiabatic basis

We derive a microscopic expression for the instantaneous diagonal elements of the density matrix $ρ_{nn}(t)$ in the adiabatic basis for an arbitrary time dependent process in a closed Hamiltonian system. If the initial density matrix is stationary (diagonal) then this expression contains only squares of absolute values of matrix elements of the evolution operator, which can be interpreted as transition probabilities. We then derive the microscopic expression for the heat defined as the energy generated due to transitions between instantaneous energy levels. If the initial density matrix is pas…

熵及其产生的比较微观研究

Comparative Microscopic Study of Entropies and their Production

本研究针对两系统能量交换的非平衡设置，通过随机矩阵理论和薛定谔方程数值积分，比较11种微观熵和3种微观温度定义的时间演化，揭示其在不同初始态和系统类型下的行为差异及热力学第二定律的违反情况。

The Second Law of Thermodynamics under Unitary Evolution and External Operations

A microscopic definition of the thermodynamic entropy in an isolated quantum system must satisfy (i) additivity, (ii) extensivity and (iii) the second law of thermodynamics. We show that the diagonal entropy, which is the Shannon entropy in the energy eigenbasis at each instant of time, meets the first two requirements and that the third requirement is satisfied if an arbitrary external operation is performed at typical times. In terms of the diagonal entropy, thermodynamic irreversibility follows from the facts that the Hamiltonian dynamics restricts quantum trajectories under unitary evoluti…

Microscopic statistical basis of classical Thermodynamics of finite systems

Heat can flow from cold to hot at any phase separation. Therefore Lynden-Bell's gravo-thermal catastrophe must be reconsidered. The original objects of Thermodynamics, the separation of phases at first order phase transitions, like boiling water in steam engines, are not described by a single canonical ensemble. Inter-phase fluctuations are not covered. The basic principles of statistical mechanics, especially of phase transitions have to be reconsidered without the use of the thermodynamic limit. Then thermo-statistics applies also to nuclei and large astronomical systems. A lot of similarity…

Density-functional treatment of model Hamiltonians: basic concepts and application to the Heisenberg model

We describe how density-functional theory, well-known for its many uses in ab initio calculations of electronic structure, can be used to study the ground state of inhomogeneous model Hamiltonians. The basic ideas and concepts are discussed for the particular case of the Heisenberg model. As representative applications, illustrating scope and limitations of the procedure, we calculate the ground-state energy of one-, two- and three-dimensional antiferromagnetic Heisenberg models in the presence of boundaries and of impurities in the bulk and at the surfaces. Correlations are shown to lift dege…