The decrease in entropy when passing from an equilibrium thermodynamic system to a slightly nonequilibrium system is investigated. A quasi-equilibrium Boltzmann distribution is used to prove the conservation of free energy during this passage. Results are obtained for a Brownian particle in a potential well with a low escape probability. The escape is interpreted as a measurement. It is shown that because of the measurement itself, the distribution function is narrower than that for a system undisturbed by measurement, i.e., an equilibrium system. In this case, the entropy difference between the equilibrium and measurement-disturbed systems is equal to the amount of information entered into the system.
The decrease in entropy when passing from an equilibrium thermodynamic system to a slightly nonequilibrium system is investigated. A quasi-equilibrium Boltzmann distribution is used to prove the conservation of free energy during this passage. Results are obtained for a Brownian particle in a potential well with a low escape probability. The escape is interpreted as a measurement. It is shown that because of the measurement itself, the distribution function is narrower than that for a system undisturbed by measurement, i.e., an equilibrium system. In this case, the entropy difference between the equilibrium and measurement-disturbed systems is equal to the amount of information entered into the system.
It is pointed out that the quantum-mechanical Boltzmann H theorem in the restricted sense (i.e. with explicit consideration of the statistics of the particles constituting the system and their interactions) does not lead directly to the distribution laws for the particles among their various states as is commonly believed, but to the equality of the probability for all states belonging to the system as a whole if the system is isolated, (or to the Boltzmann factor for the probabilities of states belonging to a system at constant temperature), in agreement with the result of the generalized quantum-mechanical H theorem as given by Pauli. The distribution laws for the particles among their various states may of course be deduced from the latter.In the course of proof, we have also introduced the following minor improvements:(i) To avoid conservation of energy in microscopic processes which lead to changes in the states of the particles and thus to equilibrium, we let our system be in interaction with a heat reservoir, so that energy may be supplied by or given to it. This is tauto-mount to changing the study of the entropy for an isolated system to that of the free energy for a system at constant temperature.(ii) Interaction energies between the particles which affect the total energy of the system and thus the distribution laws for the particles among their various states are included.
Concepts from information theory are used to investigate the relation between measurement and irreversibility in finite, closed classical systems having known Hamiltonians. The effects of measurement upon the predictions of the observer are found for the cases of a single measurement, two simultaneous measurements, and two measurements at different times. The physical basis of coarse graining and the significance of the increase in the coarse-grained entropy are discussed. An evolution equation is obtained describing the probability distribution of the measured quantities, and it is shown that this equation is non-Markoffian. Despite this non-Markoffian behavior, the distribution may behave irreversibly and may approach equilibrium. Thus, irreversible behavior is obtained without continual coarse graining. The condition for irreversible behavior is given for the case of a single measurement. The thermodynamic entropy is defined for general nonequilibrium situations.
Entropy in nonequilibrium statistical mechanics is investigated theoretically so as to extend the well-established equilibrium framework to open nonequilibrium systems. We first derive a microscopic expression of nonequilibrium entropy for an assembly of identical bosons/fermions interacting via a two-body potential. This is performed by starting from the Dyson equation on the Keldysh contour and following closely the procedure of Ivanov, Knoll and Voskresensky [Nucl. Phys. A {\bf 672} (2000) 313]. The obtained expression is identical in form with an exact expression of equilibrium entropy and obeys an equation of motion which satisfies the H-theorem in a limiting case. Thus, entropy can be defined unambiguously in nonequilibrium systems so as to embrace equilibrium statistical mechanics. This expression, however, differs from the one obtained by Ivanov {\em et al}., and we show explicitly that their ``memory corrections'' are not necessary. Based on our expression of nonequilibrium entropy, we then propose the following principle of maximum entropy for nonequilibrium steady states: ``The state which is realized most probably among possible steady states without time evolution is the one that makes entropy maximum as a function of mechanical variables, such as the total particle number, energy, momentum, energy flux, etc.'' During the course of the study, we also develop a compact real-time perturbation expansion in terms of the matrix Keldysh Green's function.
In this contribution we prove that the entropy of an N-body isolated system can not decrease and the entropy production should be non-negative provided the system possesses an equilibrium state. We define the entropy as a functional of the set of n-particle reduced density operators (n=0,....,N) generalizing the von Neumann fine-grained entropy formula. Additionally, as a consequence of our analysis we find the expression of the equilibrium n-particle reduced density operators which enter the definition of the entropy as well as the dissipated energy in an irreversible process.
We present a scheme to evaluate thermodynamic variables for a system coupled to a heat bath under a time-dependent external force using the quasi-static Helmholtz energy from the numerically "exact" hierarchical equations of motion (HEOM). We computed the entropy produced by a spin system strongly coupled to a non-Markovian heat bath for various temperatures. We showed that when changes to the external perturbation occurred sufficiently slowly, the system always reached thermal equilibrium. Thus, we calculated the Boltzmann entropy and the von Neumann entropy for an isothermal process, as well as various thermodynamic variables, such as changes in internal energies, heat, and work, for a system in quasi-static equilibrium based on the HEOM. We found that although the characteristic features of the system entropies in the Boltzmann and von Neumann cases as a function of the system-bath coupling strength are similar, those for the total entropy production are completely different. The total entropy production in the Boltzmann case is always positive, whereas that in the von Neumann case becomes negative if we chose a thermal equilibrium state of the total system (an unfactorized thermal equilibrium state) as the initial state. This is because the total entropy production in the von Neumann case does not properly take into account the contribution of the entropy from the system-bath interaction. Thus, the Boltzmann entropy must be used to investigate entropy production in the fully quantum regime. Finally, we examined the applicability of the Jarzynski equality.
We consider the temporal evolution of an isolated system of particles from a non-equilibrium state of entropy S = S′ to the equilibrium state of maximum entropy S = S max, S′ ≤ S max. The application of usual density matrix theory to the temporal development of S leads us to the unexpected result P = dS/dt = 0. The entropy does not change in time. This result is valid in case of the equilibrium state. However it is wrong for a non-equilibrium state, which could not irreversibly change from S′ to S max by entropy production P > 0. This contradicts the second law of thermodynamics. The paradoxial result can be traced back to the Liouville theorem. To overcome this difficulty, we introduce the t–P space. The time t and the entropy production P are observables in quantum thermodynamics. They are defined by the eigenvalue equations t|t〉 = t|t〉 and P|P〉 = P|P〉. The operators t and P do not commute. We have a non-vanishing commutator, [t,P] = ik, and consequently a t–P uncertainty relation, ΔtΔP ≥ k/2. The Boltzmann constant k is characteristic of quantum thermodynamics and represents the atomic entropy unit. — The time evolution operator U(τ) = exp[−(i/k)τP] shifts the time t by τ, U(τ)|t〉 = |t + τ〉. The generator of τ is the operator P of entropy production. Then the eigenvector |P〉 is also an eigenket of U(t), and it follows with progressing time τ ≥ 0 that τP ≥ 0 and hence P = dS/dt ≥ 0. This is the second law of thermodynamics, which is correctly reproduced in quantum thermodynamics. — Exchanging −τ for +τ replaces the equations of forward time +τ by the equations of backward time −τ. Independent of the direction of the arrow of time, the entropy of the system always increases or remains constant in the limit of the equilibrium state, dS ≥ 0. This is characteristic of irreversibility, for the system does not return to its initial state on time reversal. — The t–P uncertainty relation allows to discuss the temporal evolution of the expectation value 〈P〉 of entropy production and the lifetime Δt of the transient states, the system passes through on approaching the equilibrium state. In case of a non-equilibrium state 〈P〉, ΔP, and Δt are finite. On approaching the equilibrium state, 〈P〉 and ΔP decrease, consequently Δt increases. Finally, in the case of equilibrium (eq), we have 〈P〉eq = 0, ΔP eq = 0, and Δt eq → ∞. The lifetime of the equilibrium state is unlimited, because the entropy production 〈P〉 and the uncertainty of entropy production ΔP exactly vanish. The temporal evolution of the system can be described by means of a phenomenological equation.