Mathematical Physics (MP)
http://hdl.handle.net/20.500.11824/16
Mon, 25 Oct 2021 06:32:12 GMT2021-10-25T06:32:12ZAnother Proof of Born's Rule on Arbitrary Cauchy Surfaces
http://hdl.handle.net/20.500.11824/1352
Another Proof of Born's Rule on Arbitrary Cauchy Surfaces
LIll, S.; Tumulka, R.
In 2017, Lienert and Tumulka proved Born's rule on arbitrary Cauchy surfaces in Minkowski space-time assuming Born's rule and a corresponding collapse rule on horizontal surfaces relative to a fixed Lorentz frame, as well as a given unitary time evolution between any two Cauchy surfaces, satisfying that there is no interaction faster than light and no propagation faster than light. Here, we prove Born's rule on arbitrary Cauchy surfaces from a different, but equally reasonable, set of assumptions. The conclusion is that if detectors are placed along any Cauchy surface $\Sigma$, then the observed particle configuration on $\Sigma$ is a random variable with distribution density $|\Psi_\Sigma|^2$, suitably understood. The main different assumption is that the Born and collapse rules hold on any spacelike hyperplane, i.e., at any time coordinate in any Lorentz frame. Heuristically, this follows if the dynamics of the detectors is Lorentz invariant.
Thu, 14 Oct 2021 00:00:00 GMThttp://hdl.handle.net/20.500.11824/13522021-10-14T00:00:00ZFractional Diffusion and Medium Heterogeneity: The Case of the Continuos Time Random Walk
http://hdl.handle.net/20.500.11824/1333
Fractional Diffusion and Medium Heterogeneity: The Case of the Continuos Time Random Walk
Sposini, V.; Vitali, S.; Paradisi, P.; Pagnini, G.
In this contribution we show that fractional diffusion emerges from a simple Markovian Gaussian random walk when the medium displays a power-law heterogeneity. Within the framework of the continuous time random walk, the heterogeneity of the medium is represented by the selection, at any jump, of a different time-scale for an exponential survival probability. The resulting process is a non-Markovian non-Gaussian random walk. In particular, for a power-law distribution of the time-scales, the resulting random walk corresponds to a time-fractional diffusion process. We relates the power-law of the medium heterogeneity to the fractional order of the diffusion. This relation provides an interpretation and an estimation of the fractional order of derivation in terms of environment heterogeneity. The results are supported by simulations.
Sat, 24 Jul 2021 00:00:00 GMThttp://hdl.handle.net/20.500.11824/13332021-07-24T00:00:00ZStochastic Properties of Colliding Hard Spheres in a Non-equilibrium Thermal Bath
http://hdl.handle.net/20.500.11824/1332
Stochastic Properties of Colliding Hard Spheres in a Non-equilibrium Thermal Bath
Bazzani, A.; Vitali, S.; Montanari, C.; Monti, M.; Rambaldi, S.; Castellani, G.
We consider the problem of describing the dynamics of a test particle moving in a thermal bath using the stochastic differential equations. We briefly recall the stochastic approach to the Brownian based on the statistical properties of collision theory for a gas of elastic particles and the molecular chaos hypothesis. The mathematical formulation of the Brownian motion leads to the formulation of the Ornstein-Uhlenbeck equation that provides a stationary solution consistent with the Maxwell-Boltzmann distribution. According to the stochastic thermodynamics, we assume that the stochastic differential equations allow to describe the transient states of the test particle dynamics in a thermal bath and it extends their application to the study of the non-equilibrium statistical physics. Then we consider the problem of the dynamics of a test massive particle in a non homogeneous thermal bath where a gradient of temperature is present. We discuss as the existence of a local thermodynamics equilibrium is consistent with a Stratonovich interpretation of the stochastic differential equations with a multiplicative noise. The stochastic model applied to the test particle dynamics implies the existence of a long transient state during which the particle shows a net drift toward the cold region of the system. This effect recalls the thermophoresis phenomenon performed by large molecule in a solution in response to a macroscopic temperature gradient and it can be explained as an effect of the non-locality character of the collision interactions between the test particle and the thermal bath particles. To validate the stochastic model assumptions we analyze the simulation results of the 2-dimensional hard sphere gas obtained by using an event-based computer code, that solves exactly the sphere dynamics. The temperature gradient is simulated by the presence of two reflecting boundary conditions at different temperature. The simulations suggest that existence of a local thermodynamic equilibrium is justified and highlight the presence of a drift in the average dynamics of an ensemble of massive particles. The results of the paper could be relevant for the applications of stochastic dynamical systems to the non-equilibrium statistical physics that is a key issue for the Complex Systems Physics.
Sat, 24 Jul 2021 00:00:00 GMThttp://hdl.handle.net/20.500.11824/13322021-07-24T00:00:00ZImage Milnor Number Formulas for Weighted-Homogeneous Map-Germs
http://hdl.handle.net/20.500.11824/1317
Image Milnor Number Formulas for Weighted-Homogeneous Map-Germs
Pallarés, Irma; Peñafort Sanchis, Guillermo
We give formulas for the image Milnor number of a weighted-homogeneous map-germ $(\mathbb{C}^n,0)\to(\mathbb{C}^{n+1},0)$, for $n=4$ and $5$, in terms of weights and degrees. Our expressions are obtained by a purely interpolative method, applied to a result by Ohmoto. We use our approach to recover the formulas for $n=2$ and $3$ due to Mond and Ohmoto, respectively. For $n\geq 6$, the method is valid as long as certain multi-singularity conjecture holds.
Mon, 05 Jul 2021 00:00:00 GMThttp://hdl.handle.net/20.500.11824/13172021-07-05T00:00:00Z