Both quantum mechanics and classical statistical mechanics predict the same momentum and spatial density forms forms for the solution of an oscillator. In the case of quantum mechanics, the ground state is considered. One may ask whether this is a coincidence, and if not, why two seemingly different theories should lead to the same result for the oscillator case. We argue in this note that it is the form of the conservation equation of the oscillator which leads to x and p having the same probability distribution.Furthermore, given that both quantum mechanics and classical statistical mechanics are of a statistical nature, one may inquire into the actual nature of the statistics being used, the meaning behind averages being used. For example, the classical deterministic solution of an oscillator can be written in statistical terms by removing time dependence and using density = dx/v. This does not mean, however, that the actual physical motion is no longer deterministic. In addition, we try to argue that quantum mechanics may be a more detailed statistical model including dynamics which satisfies the more static statistical result of spatial density for classical statistical mechanics.Both quantum mechanics and classical statistical mechanics predict the same momentum and spatial density forms forms for the solution of an oscillator. In the case of quantum mechanics, the ground state is considered. One may ask whether this is a coincidence, and if not, why two seemingly different theories should lead to the same result for the oscillator case. We argue in this note that it is the form of the conservation equation of the oscillator which leads to x and p having the same probability distribution.Furthermore, given that both quantum mechanics and classical statistical mechanics are of a statistical nature, one may inquire into the actual nature of the statistics being used, the meaning behind averages being used. For example, the classical deterministic solution of an oscillator can be written in statistical terms by removing time dependence and using density = dx/v. This does not mean, however, that the actual physical motion is no longer deterministic. In addition, we try to argue that quantum mechanics may be a more detailed statistical model including dynamics which satisfies the more static statistical result of spatial density for classical statistical mechanics.

Dec 13, 20180 citations

Probability and Typicality in Statistical Mechanics

Barry Loewer

Fundamental theories of physicsJan 1, 20241 citation

Information Theory and Statistical Mechanics. II

E. T. Jaynes

Treatment of the predictive aspect of statistical mechanics as a form of statistical inference is extended to the density-matrix formalism and applied to a discussion of the relation between irreversibility and information loss. A principle of "statistical complementarity" is pointed out, according to which the empirically verifiable probabilities of statistical mechanics necessarily correspond to incomplete predictions. A preliminary discussion is given of the second law of thermodynamics and of a certain class of irreversible processes, in an approximation equivalent to that of the semiclassical theory of radiation.It is shown that a density matrix does not in general contain all the information about a system that is relevant for predicting its behavior. In the case of a system perturbed by random fluctuating fields, the density matrix cannot satisfy any differential equation because $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\rho}}(t)$ does not depend only on $\ensuremath{\rho}(t)$, but also on past conditions The rigorous theory involves stochastic equations in the type $\ensuremath{\rho}(t)=\mathcal{G}(t, 0)\ensuremath{\rho}(0)$, where the operator $\mathcal{G}$ is a functional of conditions during the entire interval ($0\ensuremath{\rightarrow}t$). Therefore a general theory of irreversible processes cannot be based on differential rate equations corresponding to time-proportional transition probabilities. However, such equations often represent useful approximations.Treatment of the predictive aspect of statistical mechanics as a form of statistical inference is extended to the density-matrix formalism and applied to a discussion of the relation between irreversibility and information loss. A principle of "statistical complementarity" is pointed out, according to which the empirically verifiable probabilities of statistical mechanics necessarily correspond to incomplete predictions. A preliminary discussion is given of the second law of thermodynamics and of a certain class of irreversible processes, in an approximation equivalent to that of the semiclassical theory of radiation.It is shown that a density matrix does not in general contain all the information about a system that is relevant for predicting its behavior. In the case of a system perturbed by random fluctuating fields, the density matrix cannot satisfy any differential equation because $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\rho}}(t)$ does not depend only on $\ensuremath{\rho}(t)$, but also on past conditions The rigorous theory involves stochastic equations in the type $\ensuremath{\rho}(t)=\mathcal{G}(t, 0)\ensuremath{\rho}(0)$, where the operator $\mathcal{G}$ is a functional of conditions during the entire interval ($0\ensuremath{\rightarrow}t$). Therefore a general theory of irreversible processes cannot be based on differential rate equations corresponding to time-proportional transition probabilities. However, such equations often represent useful approximations.

Physical ReviewOct 15, 19573,251 citations

TIME-RELATED ISSUES IN STATISTICAL MECHANICS

Lawrence S. Schulman

Jan 1, 20064 citations

Probability Representation of Quantum Mechanics Where System States Are Identified with Probability Distributions

Vladimir N. Chernega, Olga V. Man’ko, Vladimir I. Man’ko

The probability representation of quantum mechanics where the system states are identified with fair probability distributions is reviewed for systems with continuous variables (the example of the oscillator) and discrete variables (the example of the qubit). The relation for the evolution of the probability distributions which determine quantum states with the Feynman path integral is found. The time-dependent phase of the wave function is related to the time-dependent probability distribution which determines the density matrix. The formal classical-like random variables associated with quantum observables for qubit systems are considered, and the connection of the statistics of the quantum observables with the classical statistics of the random variables is discussed.The probability representation of quantum mechanics where the system states are identified with fair probability distributions is reviewed for systems with continuous variables (the example of the oscillator) and discrete variables (the example of the qubit). The relation for the evolution of the probability distributions which determine quantum states with the Feynman path integral is found. The time-dependent phase of the wave function is related to the time-dependent probability distribution which determines the density matrix. The formal classical-like random variables associated with quantum observables for qubit systems are considered, and the connection of the statistics of the quantum observables with the classical statistics of the random variables is discussed.

Quantum ReportsJan 21, 20205 citations

\mathcal{G}

is a functional of conditions during the entire interval ($0\ensuremath{\rightarrow}t$). Therefore a general theory of irreversible processes cannot be based on differential rate equations corresponding to time-proportional transition probabilities. However, such equations often represent useful approximations.