The statistical problem of the distribution of $s$ quanta of equal energy $ε_0$ and total energy $E$ among $N$ distinguishable particles is resolved using the conventional theory based on Boltzmann's principle of equal probabilities of configurations of particles distributed among energy levels and the concept of average state. In particular, the probability that a particle is in the \k{appa}-th energy level i.e. contains \k{appa} quanta, is given by $p(κ)=\displaystyle \frac{\displaystyle \binom{N+s-κ-2}{N-2}}{\displaystyle \binom{N+s-1}{N-1}} \;\;\; ; \;\;\; κ= 0, 1, 2, \cdots, s$ In this context, the special case ( $N=4$ , $s=4$ ) presented indicates that the alternative concept of most probable state is not valid for finite values of $s$ and $N$ . In the present article we derive alternatively $p(κ)$ by distributing $s$ quanta over $N$ particles and by introducing a new principle of equal probability of quantum states, where the quanta are indistinguishable in agreement with the Bose statistics. Therefore, the analysis of the two approaches presented in this paper highlights the equivalence of quantum theory with classical statistical mechanics for the present system. At the limit $ε_{o} \rightarrow 0 $; $s \rightarrow \infty $; $s ε_{o} = E \sim$ fixed, where the energy of the particles becomes continuous, $p(κ)$ transforms to the Boltzmann law $P(ε) = \displaystyle \frac{1}{\langle ε\rangle}e^{-\fracε{\langle ε\rangle}} \;\;\; ; \;\;\; 0\leq ε< +\infty$ where $\langle ε\rangle = E/N$ . Hence, the classical principle of equal a priori probabilities for the energy of the particles leading to the above law, is justified here by quantum mechanics.The statistical problem of the distribution of $s$ quanta of equal energy $ε_0$ and total energy $E$ among $N$ distinguishable particles is resolved using the conventional theory based on Boltzmann's principle of equal probabilities of configurations of particles distributed among energy levels and the concept of average state. In particular, the probability that a particle is in the \k{appa}-th energy level i.e. contains \k{appa} quanta, is given by $p(κ)=\displaystyle \frac{\displaystyle \binom{N+s-κ-2}{N-2}}{\displaystyle \binom{N+s-1}{N-1}} \;\;\; ; \;\;\; κ= 0, 1, 2, \cdots, s$ In this context, the special case ($N=4$, $s=4$) presented indicates that the alternative concept of most probable state is not valid for finite values of $s$ and $N$. In the present article we derive alternatively $p(κ)$ by distributing $s$ quanta over $N$ particles and by introducing a new principle of equal probability of quantum states, where the quanta are indistinguishable in agreement with the Bose statistics. Therefore, the analysis of the two approaches presented in this paper highlights the equivalence of quantum theory with classical statistical mechanics for the present system. At the limit $ε_{o} \rightarrow 0 $; $s \rightarrow \infty $; $s ε_{o} = E \sim$ fixed, where the energy of the particles becomes continuous, $p(κ)$ transforms to the Boltzmann law $P(ε) = \displaystyle \frac{1}{\langle ε\rangle}e^{-\fracε{\langle ε\rangle}} \;\;\; ; \;\;\; 0\leq ε< +\infty$ where $\langle ε\rangle = E/N$. Hence, the classical principle of equal a priori probabilities for the energy of the particles leading to the above law, is justified here by quantum mechanics.

arXiv:2111.09246Nov 17, 20210 citations

From Waves to Particles

Essential textbooks in physicsNov 1, 20180 citations

On breaking quanta

Прохоров Лев Васильевич

The nature of particles, wave functions, and the possibility of breaking quanta are discussed.

Physics of Particles and Nuclei LettersJan 1, 20091 citation

No classical particle limit for massless quanta

Riccardo Falcone, Simon Fuchs

We investigate whether relativistic massless classical particles may emerge as the classical limit of massless quanta. To address this question independently of any specific dynamics, environment, or pointer basis, we develop an axiomatic and purely kinematical framework for the coarse-graining approach. In this formulation, a candidate classical phase space is taken as the outcome space of a POVM subject only to minimal classicality and covariance under the relevant spacetime symmetry group. Applying this framework to the Poincaré group, we prove a no-go theorem for massless particles: the covariance requirement is incompatible with the operational conditions for classicality. The theorem leaves open field-like limits of massless quanta, for example the emergence of electromagnetic or gravitational fields, while ruling out classical massless particles, such as classical photons or gravitons.We investigate whether relativistic massless classical particles may emerge as the classical limit of massless quanta. To address this question independently of any specific dynamics, environment, or pointer basis, we develop an axiomatic and purely kinematical framework for the coarse-graining approach. In this formulation, a candidate classical phase space is taken as the outcome space of a POVM subject only to minimal classicality and covariance under the relevant spacetime symmetry group. Applying this framework to the Poincaré group, we prove a no-go theorem for massless particles: the covariance requirement is incompatible with the operational conditions for classicality. The theorem leaves open field-like limits of massless quanta, for example the emergence of electromagnetic or gravitational fields, while ruling out classical massless particles, such as classical photons or gravitons.

quant-pharXiv:2606.14632Jun 15, 20260 citations

From Data to Quanta

Slobodan Perović

University of Chicago Press eBooksJan 1, 20214 citations

Particles and Waves

Giulio Armando Ottonello

Springer geochemistryJan 1, 20240 citations

The statistical problem of the distribution of

s

quanta of equal energy

ε_0

and total energy

E

among

N

distinguishable particles is resolved using the conventional theory based on Boltzmann's principle of equal probabilities of configurations of particles distributed among energy levels and the concept of average state. In particular, the probability that a particle is in the \k{appa}-th energy level i.e. contains \k{appa} quanta, is given by

p(κ)=\displaystyle \frac{\displaystyle \binom{N+s-κ-2}{N-2}}{\displaystyle \binom{N+s-1}{N-1}} \;\;\; ; \;\;\; κ= 0, 1, 2, \cdots, s

In this context, the special case (

N=4

s=4

) presented indicates that the alternative concept of most probable state is not valid for finite values of

s

and

N

. In the present article we derive alternatively

p(κ)

by distributing

s

quanta over

N

particles and by introducing a new principle of equal probability of quantum states, where the quanta are indistinguishable in agreement with the Bose statistics. Therefore, the analysis of the two approaches presented in this paper highlights the equivalence of quantum theory with classical statistical mechanics for the present system. At the limit $ε_{o} \rightarrow 0 $; $s \rightarrow \infty $;

s ε_{o} = E \sim

fixed, where the energy of the particles becomes continuous,

p(κ)

transforms to the Boltzmann law

P(ε) = \displaystyle \frac{1}{\langle ε\rangle}e^{-\fracε{\langle ε\rangle}} \;\;\; ; \;\;\; 0\leq ε< +\infty

where

\langle ε\rangle = E/N

. Hence, the classical principle of equal a priori probabilities for the energy of the particles leading to the above law, is justified here by quantum mechanics.