arXiv subjects

C. Vignat

Publications and source records attributed to C. Vignat.

57 records · Page 4Linked to original sources

arXiv2005

The $p-$sphere and the geometric substratum of power law probability distributions

Links between power law probability distributions and marginal distributions of uniform laws on $p$-spheres in $\mathbb{R}^{n}$ show that a mathematical derivation of the Boltzmann-Gibbs distribution necessarily passes through power law ones. Results are also given that link parameters $p$ and $n$ to the value of the non-extensivity parameter $q$ that characterizes these power laws in the context of non-extensive statistics.

cond-mat.stat-mech↗

arXiv2005

Stochastic invertible mappings between power law and Gaussian probability distributions

We construct "stochastic mappings" between power law probability distributions (PD's) and Gaussian ones. To a given vector $N$, Gaussian distributed (respectively $Z$, exponentially distributed), one can associate a vector $X$, "power law distributed", by multiplying $X$ by a random scalar variable $a$, $N= a X$. This mapping is "invertible": one can go via multiplication by another random variable $b$ from $X$ to $N$ (resp. from $X$ to $Z$), i.e., $X=b N$ (resp. $X=b Z$). Note that all the above equalities mean "is distributed as". As an application of this stochastic mapping we revisit the so-called "zero-th law of thermodynamics problem" that bedevils the practitioners of nonextensive thermostatistics.

cond-mat.stat-mech↗

arXiv2005

Geometric origin of probabilistic distributions in statistical mechanics

A finite dimensional-system whose physics is governed by a Gaussian distribution can be regarded as a subsystem of an infinite dimensional-underlying system described by a uniform distribution on the (infinite dimensional) sphere. In turn, a finite dimensional-system ruled by a power-law probability distribution can be considered as a subsystem of a finite underlying system whose physics is governed by a uniform distribution on a finite dimensional sphere. These two uniform distributions (on finite and infinite dimensional spheres) are the most "natural" distributions since they maximize both Tsallis and Shannon entropies without any extra constraint but bounded support. The main advantage of power-law distributions could thus be assigned to the fact that they correspond to a still more "natural" underlying model, namely a finite dimensional sphere. In many physical situations, the underlying system is so large that the distinction between finite and infinite dimension is difficult to see, so that the Boltzmann-Gibbs distribution acquires a great degree of versatility. Moreover, an explicit value of the power-law exponent can be given in terms of the dimension of the underlying model and a geometric interpretation of the temperature-concept is provided.

cond-mat.stat-mech↗