In this second part of the paper, the quantum-mechanical equation of change with time is applied to an ensemble, and the quantum-mechanical analog of the classical Liouville theorem is derived. The conflict between the quantum-mechanical equation of change of an isolated system, and the second law of thermodynamics is explained. The connection between entropy and the uncertainty principle and the effect of measurement on entropy are discussed in detail. It is pointed out that in quantum mechanics, the uncertainty principle forbids the complete specification of the boundaries of a truly isolated system. The effects of random disturbances at the boundary, within the limits permitted by the uncertainty principle, are shown to cause a rapid spontaneous approach to maximum entropy of any “isolated” system. Applications to quasi-stationary irreversible effects are briefly discussed, and it is indicated how the concept of random fluctuations of the boundary leads to a deduction of the Fokker-Planck equation for irreversible processes.
In this second part of the paper, the quantum-mechanical equation of change with time is applied to an ensemble, and the quantum-mechanical analog of the classical Liouville theorem is derived. The conflict between the quantum-mechanical equation of change of an isolated system, and the second law of thermodynamics is explained. The connection between entropy and the uncertainty principle and the effect of measurement on entropy are discussed in detail. It is pointed out that in quantum mechanics, the uncertainty principle forbids the complete specification of the boundaries of a truly isolated system. The effects of random disturbances at the boundary, within the limits permitted by the uncertainty principle, are shown to cause a rapid spontaneous approach to maximum entropy of any “isolated” system. Applications to quasi-stationary irreversible effects are briefly discussed, and it is indicated how the concept of random fluctuations of the boundary leads to a deduction of the Fokker-Planck equation for irreversible processes.
This paper attempts to explain both the basic concepts and the mathematical formalism of von Neumann's operator method in quantum statistics, in terms that should be understandable by anyone who has mastered first courses in quantum mechanics and statistical mechanics. A simple model is used throughout the paper to illustrate the various theorems and techniques. The discussion begins with the statistical interpretation of the Heisenberg uncertainty principle, the definition of mixed ensembles, and the meaning of measurement in quantum mechanics. The von Neumann projection operator is explained, and is used to set up the ensemble density matrix and the von Neumann entropy operator. In the second part of the paper, to be published later, the quantum-mechanical equation of change with time is applied to the von Neumann operators.
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.
Abstract We consider the temporal evolution of an isolated system of N 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 dS/dt = 0: The entropy S does not change in time t. Thereby it is irrelevant, whether we consider a non-equilibrium state or an equilibrium state. Consequently, the system cannot irreversibly change by entropy production dS/dt > 0 from S′ to S max. This is a paradoxial result, which contradicts the experience. It can be traced back to the von Neumann equation, which in principle describes reversible processes and hence is unsuitable for calculating the irreversible evolution of the entropy S in time t. Each irreversible process is accompanied by a positive entropy production P = dS/dt ≥ 0 inside the system, which only vanishes in case of the equilibrium state. In order to overcome the above mentioned difficulties, we assign an operator P to the entropy production P, which is defined by the eigenvalue equation P |u〉 = P|u〉 with the state vector |u〉 of the N-particle system. There was an extensive discussion about the relation between the production of entropy P on one hand and the progression of time t on the other. Making use of this concept, we combine the operator of entropy production P with the time-development operator U (t, t 0) of the system and finally deduce the infinitesimal unitary operator U (t+τ,t) = 1 + (i/k)τ P by means of very general assumptions. Here P means the generator of the infinitesimal progression of time τ = t − t 0 and k the Boltzmann constant, representing the atomic entropy unit. Similarly to P we also treat the time t as an observable, defined by t |u〉 = t|u〉. We apply the infinitesimal time-evolution operator U (t+τ,t) to the operator of time t and finally obtain the P – t commutation relation i[ P , t ] = k, which is independent of τ. It shows that the operators P and t do not commute, and hence P and t are not sharply defined simultaneously. Instead we have uncertainties ΔP and Δt on measuring P and t, which are given by the P–t uncertainty relation ΔPΔt ≥ k/2. It readily allows a discussion of the evolution of the entropy S of the isolated system in time t from S′ to S max. Now, the irreversible steps are correctly described by the entropy production P = dS/dt > 0, while the thermal equilibrium is given by P = 0, ΔP = 0, and thus the lifetime of the equilibrium state Δt = ∞. According to the P–t uncertainty relation, the Boltzmann constant k is similarly important to the quantum thermodynamics of irreversible processes like Planck′s constant h to usual quantum mechanics.
To eliminate exceptions to the second law due to fluctuation phenomena, etc., the following changes are necessary. First we must state the laws of thermodynamics in terms of Gibbsian ensembles rather than of individual systems. Second we must sharpen our conception of the experimental procedure required for producing states of thermodynamic equilibrium. It will not suffice to isolate the system and wait. We must have the positive stirring action of thermal contact with a thermostat. Using the new definition of thermal equilibrium we can place statistical mechanics on a more satisfactory footing by showing that the thermostat experimental procedure leads to the canonical ensemble and maximizes v. Neumann's microscopic entropy. (Details of this proof appear in succeeding paper). Difficulties with his macroscopic definition of entropy are avoided by showing that his original microscopic definition is satisfactory when used in conjunction with a proper operational definition of thermodynamic equilibrium. In conclusion the second law of thermodynamics is derived from quantum statistics on the basis of the microscopic definition of entropy.
The second law of thermodynamics states that the entropy of an isolated system can only increase over time. This appears to conflict with the reversible evolution of isolated quantum systems under the Schrödinger equation, which preserves the von Neumann entropy. Nonetheless, one finds that with respect to many observables, expectation values approach a fixed value—their equilibrium value. This ultimately raises the question: ? For classical systems, one introduces the assumption of a low-entropy initial state along with the concept of ignorance about the microscopic details of the physical system, leading to a statistical interpretation of the second law. By considering the observables through which we examine quantum systems, both these assumptions can be incorporated, building upon recent studies of the of observables. While the statistical behavior of observable expectation values is well established, a quantitative connection to entropy increase has been lacking so far. In deriving novel bounds for the equilibration of observables, and considering the entropy of the system relative to observables, we recover a variant of the second law: the entropy with respect to a given observable tends toward its equilibrium value in the course of the system’s unitary evolution. These results also support recent findings that question the necessity of nonintegrability for equilibration in quantum systems. We further illustrate our bounds using numerical results from the paradigmatic example of a quantum Ising model on a chain of spins. There, we observe entropy increasing up to equilibrium values, as well as fluctuations, which expose the underlying reversible evolution in accordance with the derived bounds.
A general quantum theory encompassing Mechanics, Thermodynamics and irreversible dynamics is presented in two parts. The first part is concerned exclusively with the description of the states of any individual physical system. It is based on a new nonlinear quantum equation of motion, which reduces to the Schroedinger equation of motion of motion of conventional quantum dynamics only under special conditions. It accounts for the implications of the laws of Thermodynamics as well as for irreversible phenomena, such as the natural tendency of an isolated system to transit from any non-equilibrium state to an equilibrium state of higher entropy. Conversely, the laws of Thermodynamics and irreversibility emerge as manifestations of the fundamental quantum dynamical behaviour of the elementary constituents of any material system. We call this part Quantum Thermodynamics. The second part of the theory, which contains the first as a special case, is concerned with the description of stochastic distributions of states in an ensemble of identical physical systems each of which individually obeys the laws of Quantum Thermodynamics. It is based on a new measure-theoretic description of ensembles. It accounts unambiguously for the essential distinction between two types of uncertainties that are generally present in an ensemble, namely, quantal uncertainties due to the inherent quantal nature of the states of each individual member system and nonquantal uncertainties due to the stochastic distribution of states. We call this part Quantum Statistical Thermodynamics.