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Quantum and classical approaches in statistical physics: some basic inequalities and game theoretic interpretation

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. 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 $h{\to}0$, $T{\to}\infty$, $\omega{\to}0$, and $N{\to}\infty,$ where $h$ is the Planck constant, $T$ is the absolute temperature, $\omega$ is the frequency of an oscillator, and $N$ 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.

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The extremal principles, for instance, the Fermat's principle of least time and the principle of least action remain central in modern physics. The named above extremal principles are formulated in terms of one physical value. In the present paper, we give the game theoretic interpretation of the interaction of the three physical values: mean energy E, entropy S and free energy F. Such approach is inspired by the Neumann--Morgenstern game theory and the Nash game theory. According to these game theories, the transition from the deterministic strategy to probabilistic strategy increases the winning for all players. From this point of view we consider the connection between classical mechanics (deterministic strategy) and quantum mechanics (probabilistic strategy). In the present paper, we improve, develop and generalize our previous results.

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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.
The extremal principles, for instance, the Fermat's principle of least time and the principle of least action remain central in modern physics. The named above extremal principles are formulated in terms of one physical value. In the present paper, we give the game theoretic interpretation of the interaction of the three physical values: mean energy E, entropy S and free energy F. Such approach is inspired by the Neumann--Morgenstern game theory and the Nash game theory. According to these game theories, the transition from the deterministic strategy to probabilistic strategy increases the winning for all players. From this point of view we consider the connection between classical mechanics (deterministic strategy) and quantum mechanics (probabilistic strategy). In the present paper, we improve, develop and generalize our previous results.
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 start from the fundamental premise that any physical interaction can be interpreted as a game. To demonstrate this, we draw upon the free energy principle and the theory of quantum reference frames. In this way, we place the game-theoretic Nash Equilibrium in a new light in so far as the incompleteness and undecidability of the concept, as well as the nature of strategies in general, can be seen as the consequences of certain no-go theorems. We show that games of the generic imitation type follow a circularity of idealization that includes the good regulator theorem, generalized synchrony, and undecidability of the Turing test. We discuss Bayesian games in the light of Bell non-locality and establish the basics of quantum games, which we relate to local operations and classical communication protocols. In this light, we also review the rationality of gaming strategies from the players’ point of view.
This work concentrates on simultaneous move non-cooperating quantum games. Part of it is evidently not new, but it is included for the sake self consistence, as it is devoted to introduction of the mathematical and physical grounds of the pertinent topics, and the way in which a simple classical game is modified to become a quantum game (a procedure referred to as a quantization of a classical game). The connection between game theory and information science is briefly stressed, and the role of quantum entanglement (that plays a central role in the theory of quantum games), is exposed. Armed with these tools, we investigate some basic concepts like the existence (or absence) of a pure strategy and mixed strategy Nash equilibrium and its relation with the degree of entanglement. The main results of this work are as follows: 1) Construction of a numerical algorithm based on the method of best response functions, designed to search for pure strategy Nash equilibrium in quantum games. The formalism is based on the discretization of a continuous variable into a mesh of points, and can be applied to quantum games that are built upon two-players two-strategies classical games, based on the method of best response functions. 2) Application of this algorithm to study the question of how the existence of pure strategy Nash equilibrium is related to the degree of entanglement (specified by a continuous parameter γ ). It is shown that when the classical game GC has a pure strategy Nash equilibrium that is not Pareto efficient, then the quantum game GQ with maximal entanglement (γ = π/2) has no pure strategy Nash equilibrium. By studying a non-symmetric prisoner dilemma game, it is found that there is a critical value 0γc such that for γγc there is a pure strategy Nash equilibrium and for γ≥γc there is no pure strategy Nash equilibrium. The behavior of the two payoffs as function of γ starts at that of the classical ones at (D, D) and approaches the cooperative classical ones at (C, C) (C = confess, D = don’t confess). 3) We then study Bayesian quantum games and show that under certain conditions, there is a pure strategy Nash equilibrium in such games even when entanglement is maximal. 4) We define the basic ingredients of a quantum game based on a two-player three strategies classical game. This requires the introduction of trits (instead of bits) and quantum trits (instead of quantum bits). It is proved that in this quantum game, there is no classical commensurability in the sense that the classical strategies are not obtained as a special case of the quantum strategies.