Toward a quantum generalization of equilibrium statistical thermodynamics: ħ-k Dynamics

Abstract

We show that the quantum statistical mechanics (QSM ) describing the quantum and thermal properties of objects only has the sense of a particular semiclassical approximation. We propose a more general microdescription than in QSM of objects in a thermal bath with the vacuum explicitly taken into account; we call it -k dynamics. We construct a qualitatively new model of the object environment, namely, a quantum thermal bath. We study its properties including the cases of a "cold" and a "thermal" vacuum. We introduce the stochastic action operator and show its fundamental role in the microdescription. We establish that the corresponding macroparameter, the effective action, plays just as significant a role in the macrodescription. The most important effective macroparameters of equilibrium quantum statistical thermodynamics-internal energy, temperature, and entropy-are expressed in terms of this macroparameter. Keywords:-k dynamics, quantum thermal bath, cold vacuum, thermal vacuum, stochastic action operator, effective action, quantum-thermal entropy Possible approaches to a consistent quantum-thermal description of natural objectsThe conviction that equilibrium quantum statistical mechanics (QSM) not only is an adequate description of microobjects in a thermal bath but also forms a basis for the corresponding macrodescription has predominated for a fairly long time. In other words, it is assumed that QSM allows obtaining thermodynamic macroparameters from the microdescription and establishing observable interrelations between them (thermodynamic laws, equations of state, etc.) [1]. At the same time, it is well known that there exist such macroparameters, temperature for example, whose analogues have not yet been studied on the microlevel. This leads to introducing the idea of the independence and equality of the micro-and macrodescriptions of nature with a certain interrelation between them [2], [3].We briefly analyze already revealed elements of the limitations of equilibrium QSM as a theory claiming to consistently describe all quantum and thermal phenomena. As is well known, QSM is based on the notion of the density matrix (operator), which in the energy representation has the form of the Gibbs-von Neumann quantum canonical distribution,where ε n is the spectrum of the object energy, F is the free energy determined by the normalization condition, and Θ is the inverse modulus of the distribution. It has the meaning of the Lagrange multiplier in the derivation of distribution (1) from the maximum entropy principle under thermal equilibrium conditions [4].

The Schrödinger uncertainty relation as a key tool for incorporating statistical thermodynamics into quantum theory

“…In ref. 1, we showed that, in the general case, these vacuum states are described by the family of complex wave functions that are dependent on the particular choice of parameters and .…”

“…We now discuss the invariance of the zeroth law with respect to the transitions between equilibrium thermal CCSs at nonzero temperatures. To do this, we use the invariance of the zeroth law with respect to the Lorentz group O (2,1). This can be demonstrated in the simplest way using the comment to formula (27).…”

“…Thus, the state of the cold vacuum in the group of Bogolyubov transformations corresponds to the proper inertial reference system in the special theory of relativity for the group O (1,1) in the space of states. Other inertial reference systems that are equivalent to the proper inertial reference system correspond to different states of the arbitrary vacuum.…”

“…Introducing the operators of the stochastic influence with consequent separate calculation of uncertainties product and of external action reveals the physical mechanism of the process of equilibrium establishment between an object and its environment. Thus, we arrive at the conclusion that, among so far existing theoretical approaches, only (ħ, k) dynamics [1] with temperaturedependent complex wave functions, being a modifying quantum theory, provides the possibility to include the zeroth law at nonzero temperatures in its apparatus. Of course, the possibility of constructing an adequate theory using the complex density matrix remains open.…”

Arbitrary vacuums as a model of stochastic environment: On the problem of incorporating thermodynamics into quantum theory

“…However, Umezawa recognized that he failed to solve this prob lem in the context of the thermal field dynamics developed by him. We will show in this article and sub sequent one that our approach (called (ប, k) dynam ics [2][3][4][5][6]) apparently allows us to come as close to this target as possible.…”