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arXiv:2006.11329

Quantum and classical approaches in statistical physics: some basic inequalities

Lev Sakhnovich

Published Jun 19, 2020• 0 citations
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Abstract

We present some basic inequalities between the classical and quantum values of free energy, entropy and mean energy. We investigate the transition from the deterministic case (classical mechanics) to the probabilistic case (quantum mechanics). In the first part of the paper, we assume that the reduced Planck constant $\hbar$, the absolute temperature $T$, the frequency of an oscillator $ω$, and the degree of freedom of a system $N$ are fixed. This approach to the problem of comparing quantum and classical mechanics is new (see [35]--[37]). In the second part of the paper, we simultaneously derive the semiclassical limits for four cases, that is, for $\hbar{\to}0$, $T{\to}\infty$, $ω{\to}0$, and $N{\to}\infty$. We note that only the case $\hbar{\to}0$ is usually considered in quantum mechanics (see [21]). The cases $T{\to}\infty$ and $ω{\to}0$ in quantum mechanics were initially studied by M. Planck and by A. Einstein, respectively.

DOI
10.48550/arxiv.2006.11329

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We present some basic inequalities between the classical and quantum values of free energy, entropy and mean energy. We investigate the transition from the deterministic case (classical mechanics) to the probabilistic case (quantum mechanics). In the first part of the paper, we assume that the reduced Planck constant ℏ\hbarℏ, the absolute temperature TTT, the frequency of an oscillator ωωω, and the degree of freedom of a system NNN are fixed. This approach to the problem of comparing quantum and classical mechanics is new (see [35]--[37]). In the second part of the paper, we simultaneously derive the semiclassical limits for four cases, that is, for ℏ→0\hbar{\to}0ℏ→0, T→∞T{\to}\inftyT→∞, ω→0ω{\to}0ω→0, and N→∞N{\to}\inftyN→∞. We note that only the case ℏ→0\hbar{\to}0ℏ→0 is usually considered in quantum mechanics (see [21]). The cases T→∞T{\to}\inftyT→∞ and ω→0ω{\to}0ω→0 in quantum mechanics were initially studied by M. Planck and by A. Einstein, respectively.
We present some basic inequalities between the classical and quantum values of free energy, entropy and mean energy. The extremal principles (e.g., the Fermat's principle of least time and the least action principle), which remain central in modern physics, are formulated in terms of one physical value. Using our inequalities, we give a game theoretic interpretation of the interaction of three physical values, namely, free energy, entropy and mean energy. Such approach is inspired by Neumann--Morgenstern and Nash game theories. According to these game theories, the transition from the deterministic strategy (classical mechanics) to probabilistic strategy (quantum mechanics) increases the winning for all players. In the second part of the paper, we simultaneously derive the quasi-classical limits for four cases, that is, for h→0h{\to}0h→0, T→∞T{\to}\inftyT→∞, ω→0\omega{\to}0ω→0, and N→∞,N{\to}\infty,N→∞, where hhh is the Planck constant, TTT is the absolute temperature, ω\omegaω is the frequency of an oscillator, and NNN is the degree of freedom of a system. We note that only the first case is usually considered in quantum physics. The second case was studied initially by M. Planck and the third case by A.~Einstein.
Classical statistical mechanics/thermodynamics is sometimes portrayed as an approximate approach to describing the classical mechanical motion of very large numbers of particles. Equilibrium statistical mechanics/thermodynamics, however, is a special situation in which thermodynamic averages such as temperature, pressure etc do not change in time. We argue in this note that this is due to constraints imposed on such equilibrium scenarios namely time reversal reaction balance. It is this (approximate) constraint which yields the Maxwell-Boltzmann distribution which in turn defines average values of classical values e.g. pressure resulting from an average of bvv (b=constant) and temperature being related to average kinetic energy. Interestingly, average values such as average energy and pressure may yield a classical mechanical relationship for adiabatic processes, otherwise the more complicated dE=TdS - PdV applies to these averages. (To be more accurate one may use the constraints of microcanonical theory.) Quantum mechanics is also a statistical theory because a single particle with a fixed momentum is characterized by a physical wavelength region and a probability distribution exp(ipx). If interactions occur with potential regions a superposition of these distributions (i.e. a momentum superposition) occurs i.e. W(x)= Sum over p a(p)exp(ipx). This represents one constraint in quantum mechanics), but there is a second, namely that averages for a bound system match classical values. Thus, like classical statistical mechanics one may imagine a system characterized by a parameter say L (box length) which changes such that the particle stays in the same quantum state (i.e. an adiabatic change). Given that bound quantum mechanics is constrained to behave on average as a classical system it is not surprising that a dE=PdV which is essentially a Newton’s second law statement dE=Fdx and is not necessarily linked to a usual statistical system. If a quantum state has its parameter changed nonadiabatically it should be able to have it change into a superposition of new quantum states (new L) which is very different from the classical statistical mechanical scenario. Furthermore the ideal gas relationship PV=NRT and E=3/2NRT (3 degrees of freedom) or PV=(⅔)E matches a classical mechanical result because a particle with p may bounce from a wall yielding an impulse. In this note we try to examine some of these underlying constraints and argue that even dE=PdV for average E and P values (i.e. no entropy change) one must be aware of the specific constraints involved. In other words because dE=PdV for both adiabatic classical statistical mechanical systems and a quantum eigenstate, these do not necessarily represent the same statistical type of physics. We further argue that quantum mechanical bound states are constrained to have some average and rms values such as KE(x) and p rms behave as classical values and so these should not be taken to represent “intrinsic” quantum behaviour while average spatial density on the other hand does. (It is linked to the constraint of wavelength.)
Classical statistical mechanics tries to derive the macroscopic properties of matter starting from the mechanical laws that rule the behaviour of single particles. To describe equilibrium states, only observables accounting for correlations among states are considered, and only systems consisting of a large number of particles are taken into account. The observables are described by continuous functions on phase space, and the states are represented by linear assignments of a number to each observable. Within the framework of the canonical ensemble, one deals with mechanical systems in thermal equilibrium with a thermal reservoir, and the equilibrium state is reached as a result of the interaction with the external world. The external world may be really external, or equally well, the unrecognized internal degrees of freedom, like in the calculation of viscosity or thermal conductivity. If really only the external world is the cause we may expect some surface dependence while an internal unrecognized degree of freedom would have a volume effect, unless the interactions are long range. The microcanonical ensemble is instead introduced to study isolated mechanical systems, and the equilibrium is viewed as a temporal average, rather than as a limit. Attention is then focused on partition functions, the theorem of equipartition of energy and an elementary theory of specific heats. In the second part, the Planck derivation of the law of black-body radiation is analysed, presenting in chronological order the Kirchhoff theorem, the Stefan law, the Wien displacement law, the Rayleigh–Jeans formula and the Planck hypothesis. Further topics discussed are the Einstein and Debye quantum models for specific heats of solids. These topics prepare the ground for the introduction of quantum statistical mechanics. The third part is, in fact, devoted to the analysis of identical particles in quantum mechanics.
We argue that classical statistical mechanics (say of an ideal gas) is based on a single equilibration scheme characterized by a temperature T (even in the presence of a potential V(x)). This single scheme is based on the maximization of entropy (subject to an average energy constraint) in order to determine probabilities for p,v and x. Quantum mechanics, on the other hand, is based on two equilibration schemes, we argue. The first is due to a built-in probability of exp(ipx) for a free particle. In this case, a potential V(x) interacts with momentum (impulse hits) to create a potential energy. No velocity appears in the creation of potential energy, but does appear in the creation of average kinetic energy. Like with a classical mechanical bound state, energy becomes a system variable. This is the first equilibration. We argue that it should have its own entropy. The probability form exp(ipx) together with conservation of energy and boundary conditions fix allowable single particle energy values (which are discrete) unlike in the classical statistical mechanical case. The second equilibration occurs if there exists a mechanism for moving from one discrete E to another, e.g. photon emission/absorption. This equilibration is based on maximization of entropy (a second entropy), but the effects of the first equilibration remain because it is this first scheme which controls allowable E values. We also argue that exp(-i Et) from the first equilibration scheme may affect the usual entropy based on exp(-E/T) used in the second scheme.
Classical mechanics in the ideal gas scenario and quantum mechanics make use of a set of energy levels which a single particle may occupy. Thus the problem becomes to determine the probability associated with ei i.e. P(ei). In this note we take the approach that relative probabilities depend on the energy gap in question. For example, consider a quantum two level system where ei values are provided by the time-independent Schrodinger equation. From the point of view of statistics, these are simply a set of ei values. Their associated physics is contained in the Schrodinger equation. For the quantum two level system one must find the relative probabilities PR(e1) and PR(e2). Next, imagine a quantum system with many levels and assume for the sake of argument there are levels e1,e2 and ei,e(i+1) such that e2-e1=e(i+1)-ei. If the mechanism is the same for jumping from e1 to e2 and from ei to e(i+1) (i.e. absorbing a photon of the same energy), one might argue that the relative probabilities PR(e2)/PR(e1) = PR(e(i+1))/PR(ei). From this one may obtain the Maxwell-Boltzman result exp(-ei/T)). Note that levels do not need to be adjacent. For example in a large level system one might consider e1 and e3 and any ei and ej such that e3-e1 = ej - ei. In some situations this may need to be modified (e.g. fermions and bosons have enhancements and there may be level related issues), but as a first step we argue that insisting on relative probabilities being dependent on ei-ej has important consequences. Furthermore, in such a case the normalization constant for these relative probabilities i.e. C(T)= Sum over i exp(-ei/T) is important. Taking ln(C(T) and then taking the derivative d/dT yields an expression proportional to the average energy. An expression for dE/dT dT, however, would lead to a second order differential equation in ln(C(T)). If one introduces a new function S such that E-TS = -Tln(C), then one may show that dE/dT = T dS/dT i.e. entropy S emerges.
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/2E_k=k_B T/2Ek​=kB​T/2, where kBk_BkB​ is the Boltzmann constant and TTT 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⟩E_k = \langle \mathcal E_k \rangleEk​=⟨Ek​⟩, where Ek\mathcal E_kEk​ is thermal kinetic energy per one degree of freedom of the thermostat consisting of harmonic oscillators and ⟨...⟩\langle ...\rangle⟨...⟩ denotes averaging over frequencies ω\omegaω of those thermostat oscillators which contribute to EkE_kEk​ according to the probability distribution P(ω)\mathbb P(\omega)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⟩E_k = \langle \mathcal E_k \rangleEk​=⟨Ek​⟩ for other quantum systems.