Randomness is viewed through an analogy between a physical quantity, density of gas, and a mathematical construct -- probability density. Boltzmann's deduction of equilibrium distribution of ideal gas placed in an external potential field than provides a way of viewing probability density from a perspective of forces/potentials, hidden behind it.
Randomness is viewed through an analogy between a physical quantity, density of gas, and a mathematical construct -- probability density. Boltzmann's deduction of equilibrium distribution of ideal gas placed in an external potential field than provides a way of viewing probability density from a perspective of forces/potentials, hidden behind it.
The relationship between chance and determinism has been analyzed by philosophy, mathematics, and physics.With the aim of identifying the intersection between the two, the Law of Probability Equilibrium (LPE) was demonstrated.When the probabilities of natural phenomena differ from zero and one, the LPE shows that the observed probability is empirical and that the cumulative probability functions of these phenomena measure the imminence of their occurrence ("energy concentration").These functions behave like waves that carry information about the probabilistic ("energetic") potential for the materialization of such phenomena.Probability functions exhibit properties analogous to quantum systems, such as complementarity, superposition, uncertainty, collapse, duality, interference, and entanglement.The findings indicate that chance and determinism converge at a cumulative limit of 1/2, forming a self-regulating equilibrium within the probabilistic field.This implies both the certainty of occurrence and the unpredictability of its timing.Even events with infinitesimal probabilities will inevitably occur.The LPE shows that the probability potential decreases until it reaches zero ("maximum entropy").Accumulated randomness is restricted to a maximum value of 1/2, allowing for causal processes (e.g., emission and absorption).Each probabilistic event, before being observed, exists in two superposed states: presence and absence, and it possesses a simultaneous and distinct probabilistic duality.One of these values is derived from the density function at the final instant, while the other is accumulated and remains stationary at 1/2.Upon measurement, the two probability density functions collapse into presence or absence (1 or 0) ("particle modality").Meanwhile, the two cumulative probability distribution functions remain entangled, each How to cite this paper: Traversa, I.
A fundamental postulate of statistical mechanics is that all microstates in an isolated system are equally probable. This postulate, which goes back to Boltzmann, has often been criticized for not having a clear physical foundation. In this note, we provide a derivation of the canonical (Boltzmann) distribution that avoids this postulate. In its place, we impose two axioms with physical interpretations. The first axiom (thermal equilibrium) ensures that, as our system of interest comes into contact with different heat baths, the ranking of states of the system by probability is unchanged. Physically, this axiom is a statement that in thermal equilibrium, population inversions do not arise. The second axiom (energy exchange) requires that, for any heat bath and any probability distribution on states, there is a universe consisting of a system and heat bath that can achieve this distribution. Physically, this axiom is a statement that energy flows between system and heat bath are unrestricted. We show that our two axioms identify the Boltzmann distribution.
The Gibbs entropy of a macroscopic classical system is a function of a probability distribution over phase space, i.e., of an ensemble. In contrast, the Boltzmann entropy is a function on phase space, and is thus defined for an individual system. Our aim is to discuss and compare these two notions of entropy, along with the associated ensemblist and individualist views of thermal equilibrium. Using the Gibbsian ensembles for the computation of the Gibbs entropy, the two notions yield the same (leading order) values for the entropy of a macroscopic system in thermal equilibrium. The two approaches do not, however, necessarily agree for non-equilibrium systems. For those, we argue that the Boltzmann entropy is the one that corresponds to thermodynamic entropy, in particular in connection with the second law of thermodynamics. Moreover, we describe the quantum analog of the Boltzmann entropy, and we argue that the individualist (Boltzmannian) concept of equilibrium is supported by the recent works on thermalization of closed quantum systems.
The Gibbs entropy of a macroscopic classical system is a function of a probability distribution over phase space, i.e., of an ensemble. In contrast, the Boltzmann entropy is a function on phase space, and is thus defined for an individual system. Our aim is to discuss and compare these two notions of entropy, along with the associated ensemblist and individualist views of thermal equilibrium. Using the Gibbsian ensembles for the computation of the Gibbs entropy, the two notions yield the same (leading order) values for the entropy of a macroscopic system in thermal equilibrium. The two approaches do not, however, necessarily agree for non-equilibrium systems. For those, we argue that the Boltzmann entropy is the one that corresponds to thermodynamic entropy, in particular, in connection with the second law of thermodynamics. Moreover, we describe the quantum analog of the Boltzmann entropy, and we argue that the individualist (Boltzmannian) concept of equilibrium is supported by the recent works on thermalization of closed quantum systems.
We report an inconsistency found in probability theory (also referred to as measure-theoretic probability). For probability measures induced by real-valued random variables, we deduce an "equality" such that one side of the "equality" is a probability, but the other side is not. For probability measures induced by extended random variables, we deduce an "equality" such that its two sides are unequal probabilities. The deduced expressions are erroneous only when it can be proved that measure-theoretic probability is a theory free from contradiction. However, such a proof does not exist. The inconsistency appears only in the theory rather than in the physical world, and will not affect practical applications as long as ideal events in the theory (which will not occur physically) are not mistaken for observable events in the real world. Nevertheless, unlike known paradoxes in mathematics, the inconsistency cannot be explained away and hence must be resolved. The assumption of infinite additivity in the theory is relevant to the inconsistency, and may cause confusion of ideal events and real events. As illustrated by an example in this article, since abstract properties of mathematical entities in theoretical thinking are not necessarily properties of physical quantities observed in the real world, mistaking the former for the latter may lead to misinterpreting random phenomena observed in experiments with microscopic systems. Actually the inconsistency is due to the notion of "numbers" adopted in conventional mathematics. A possible way to resolve the inconsistency is to treat "numbers" from the viewpoint of constructive mathematics.