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Quantum and classical approaches in statistical physics: game theoretic interpretation

Lev Sakhnovich

Published Jun 19, 2020• 0 citations
OpenAlex

Abstract

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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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. 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.
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.
We consider game theory from the perspective of quantum algorithms. Strategies in classical game theory are either pure (deterministic) or mixed (probabilistic). While not every two-person zero-sum finite game has an equilibrium in the set of pure strategies, von Neumann showed that there is always an equilibrium at which each player follows a mixed strategy. A mixed strategy deviating from the equilibrium strategy cannot increase a player's expected payoff. We show by example, however, that a player who implements a quantum strategy can increase his expected payoff, and explain the relation to efficient quantum algorithms.
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.
Quantum games have proposed a new point of view for the solution of the classical problems and dilemmas in game theory. Certain quantization relationships can be proposed with the objective that a game can be generalized into a quantum domain where the linear superposition of actions is allowed. This quantization let us describe and solution problems originated by conflicting or cooperative behaviors among the members of a system from the point of view of quantum mechanical interactions. This leads us to interesting aspects which only can be observed through the quantization of a game like the possibility of the entanglement between players, the definition of a socioeconomical temperature in a system, the collective welfare principle,the analysis of a game through elements of quantum information theory and the globalization process. The relationships between quantum mechanics and game theory are also studied and its shown how although both systems analyzed are described through two theories apparently different (quantum mechanics and game theory) both are analogous and thus exactly equivalents.