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.
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.
Abstract In this paper the thermodynamical properties of any system whatsoever are deduced from quantum mechanics. Two fundamental irreversible processes are considered: the conversion of other forms of energy into heat, and the flow of heat from one temperature to another. By the proof of a generalized H -theorem, it is shown that in each case the entropy, correctly defined, must increase, and the system tend towards a state of equilibrium. A simple but rigorous proof of Boltzmann’s law is given from which the thermodynamics of reversible processes may be inferred. An appendix includes the exact solution of the most general time-dependent perturbation problem of quantum mechanics.
A fundamental postulate of statistical mechanics is that all microstates in an isolated system are equally probable. This postulate, which goes back to Boltzmann, has often been criticized for not having a clear physical foundation. In this note, we provide a derivation of the canonical (Boltzmann) distribution that avoids this postulate. In its place, we impose two axioms with physical interpretations. The first axiom (thermal equilibrium) ensures that, as our system of interest comes into contact with different heat baths, the ranking of states of the system by probability is unchanged. Physically, this axiom is a statement that in thermal equilibrium, population inversions do not arise. The second axiom (energy exchange) requires that, for any heat bath and any probability distribution on states, there is a universe consisting of a system and heat bath that can achieve this distribution. Physically, this axiom is a statement that energy flows between system and heat bath are unrestricted. We show that our two axioms identify the Boltzmann distribution.
One of the fundamental laws of classical statistical physics is the energy equipartition theorem which states that for each degree of freedom the average kinetic energy equals Ek=kBT/2, where kB is the Boltzmann constant and T is temperature of the system. Despite the fact that quantum mechanics has already been developed for more than 100 years still there is no quantum counterpart of this theorem. We attempt to fill this far-reaching gap and formulate the quantum law for equipartition of energy in the appealing form Ek=⟨Ek⟩, where Ek is thermal kinetic energy per one degree of freedom of the thermostat consisting of harmonic oscillators and ⟨...⟩ denotes averaging over frequencies ω of those thermostat oscillators which contribute to Ek according to the probability distribution P(ω). We derive it for two paradigmatic and exactly solvable models of quantum open systems: a free Brownian particle and a harmonic oscillator. We formulate conditions for validity of the relation Ek=⟨Ek⟩ for other quantum systems.
The statistical problem of the distribution of s quanta of equal energy ε0 and total energy E among N distinguishable particles is resolved using the conventional theory based on Boltzmann's principle of equal probabilities of configurations of particles distributed among energy levels and the concept of average state. In particular, the probability that a particle is in the \k{appa}-th energy level i.e. contains \k{appa} quanta, is given by p(κ)=(N−1N+s−1)(N−2N+s−κ−2);κ=0,1,2,⋯,s In this context, the special case (N=4, s=4) presented indicates that the alternative concept of most probable state is not valid for finite values of s and N. In the present article we derive alternatively p(κ) by distributing s quanta over N particles and by introducing a new principle of equal probability of quantum states, where the quanta are indistinguishable in agreement with the Bose statistics. Therefore, the analysis of the two approaches presented in this paper highlights the equivalence of quantum theory with classical statistical mechanics for the present system. At the limit $ε_{o} \rightarrow 0 $; $s \rightarrow \infty $; sεo=E∼ fixed, where the energy of the particles becomes continuous, p(κ) transforms to the Boltzmann law P(ε)=⟨ε⟩1e−⟨ε⟩ε;0≤ε<+∞ where ⟨ε⟩=E/N. Hence, the classical principle of equal a priori probabilities for the energy of the particles leading to the above law, is justified here by quantum mechanics.
Classical statistical equilibrium, for a Maxwell-Boltzmann gas for example, is based on localizing the gas to a certain volume and having no average changes in time. In other words, maximization of entropy subject to the average energy constraint sum over i ei P(ei) is linked with these two ideas. Ei (kinetic energy for no V(x)) is a conserved quantity in reactions and overall energy represents a quantity which does not change in time consistent with a center-of-mass which does not change and average values which do not change in time. This approach is associated with a specific kind of statistics, namely that of the constituents of the system. As a result, one obtains the Maxwell-Boltzmann distribution exp(-ei/T) where ei is an extensive property of a particle (or subset of the system) and T an intensive property linked with the whole system. Entire systems, however, may interact. For example, a N particle system with T1 may achieve equilibrium with a N system with T2 yielding a new average (T1+T2)/2. The probabilities exp(-ei/T), however, are for subcomponents of a system. One could argue that momentum is an additive (vector) quantity which is conserved in collisions, but the average momentum of the system is 0 which does not allow one to describe increases in average p and -p which may be described using energy, i.e. through T temperature. This begs the question whether momentum may be associated with maximization of entropy subject to a constraint. In (1), we argued that there were flow probabilities in quantum mechanics linked with exp(-iEt) and exp(ipx), which may be obtained formally through maximization of entropy. Here we argue that while an MB gas describes a localized volume with average values not changing in time, x and t state values change, even if p and pp/2m remain constant for a free particle and it may be possible to have a maximum entropy situation to describe the combination of px and -Et. In other words, we suggest localized versus nonlocalized equilibria. The latter represents a quantum free particle, but one may then try to localize such a particle in a potential. What happens in such a case? One does not switch to a localized maximization of entropy as in the MB gas case, because it is possible to have the potential leave the flow unbroken as argued in (1). In other words, the nonlocalized free particle equilibrium is akin to a particle bouncing back and forth in a box, which in classical mechanics is not considered a statistical process. Here we suggest that the underlying picture is statistical meaning that the classical mechanics approach seems to be an approximation. We consider both a quantum particle trapped in an infinite potential well and the ground state of an oscillator to examine ideas of maximizing entropy beyond exp(ipx). A key common feature of both types of equilibrium is the product rule P(ei)P(ej) = P(ei+ej) or exp(ip1x)exp(ip2x) = exp(i (p1+p2)x). We consider this feature as well.
Abstract The quantum thermodynamics of the ideal gas is treated by means of the particle entropies σ=ka. The non-dimensional entropy numbers a are calculated by the extended, temperature-dependent Schrödinger equation applying periodic boundary conditions. The Boltzmann constant k represents the atomic entropy unit. The quantized particle entropies a are used to calculate the Boltzmann distribution f=exp(α−a), the density of entropy levels D(a), the dimensionless chemical potential α=μ/kT, etc. – The thermodynamics of the ideal gas is treated on the basis of entropy quanta by means of the density matrix formalism. In this way we obtain the total particle entropy A, the thermodynamic entropy S (equation of Sackur-Tetrode), the internal and the Helmholtz free energy E and F, respectively, etc. Moreover, it can also be shown on the basis of entropy quanta that the entropy of a closed system in thermal equilibrium has a maximum value, S=S max. Against this, the entropy S ′ of an arbitrary non-equilibrium state is always smaller, S′ < S max (second law of thermodynamics). – Application of ordinary density matrix theory to a closed system of N particles leads to the result that the entropy S does not change in time, dS/dt=0, regardless whether we consider an equilibrium state, S=S max, or a non-equilibrium state, S=S′. Consequently, S cannot irreversibly change from S′ to S max. However, any irreversible process is accompanied by a positive entropy production P=dS/dt>0. In order to overcome this obvious contradiction, we discuss the entropy evolution in time by means of the commutator equation G =i[ P , t ], which is deduced from very general assumptions. Here P and t are the operators of the entropy production P and the time t. Accordingly, P and t do not commute in general, and hence P and t are not sharply defined simultaneously. Instead we have uncertainties ΔP and Δt, which are expressed by the uncertainty relation of the N-particle system, ΔPΔt≥(1/2)|〈 G 〉|=(γ/2)k. This P − t uncertainty relation easily allows a discussion of the evolution of the entropy S in time t from S′ to S max. Now the irreversible steps are correctly described by the entropy production P=dS/dt > 0, and the thermal equilibrium by P=0, ∆P=0, and thus the lifetime ∆t=∞.