Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

AIP conference proceedings

Randomness as an equilibrium. Potential and probability density

Marián Grendár

Published Jan 1, 2002• 0 citations
Publisher

Abstract

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.

DOI
10.1063/1.1477062

Related papers

Randomness as an Equilibrium. Potential and Probability Density

M. Grendar, Grendar, M.

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.

arXiv:math/0212019•Dec 2, 2002•0 citations

Law of Probability Equilibrium (LPE)？and Bounded Randomness: Light in？the Universe, Photon Emission？and Black Holes

Ignacio Traversa

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.

OALib•Jan 1, 2025•0 citations

Axioms for the Boltzmann Distribution

Adam M. Brandenburger, Kai Steverson

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.

Foundations of Physics•May 1, 2019•5 citations

Gibbs and boltzmann entropy in classical and quantum mechanics

Sheldon Goldstein, Joel L. Lebowitz, Roderich Tumulka +1 more

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.

arXiv:1903.11870•Mar 28, 2019•17 citations

Gibbs and Boltzmann Entropy in Classical and Quantum Mechanics

Sheldon Goldstein, Joel L. Lebowitz, Roderich Tumulka +1 more

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.

WORLD SCIENTIFIC eBooks•Apr 21, 2020•62 citations

On randomness in quantum mechanics

Alberto C. de la Torre

The quantum mechanical probability densities are compared with the probability densities treated by the theory of random variables. The relevance of their difference for the interpretation of quantum mechanics is commented.

arXiv:0707.2954•European Journal of Physics•Apr 25, 2008•5 citations
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.