From a kinetic equation to a diffusion under an anomalous scaling

A linear Boltzmann equation is interpreted as the forward equation for the probability density of a Markov process (K(t), i(t), Y(t)), where (K(t), i(t)) is an autonomous reversible jump process, with waiting times between two jumps with finite expectation value but infinite variance, and Y(t) is an additive functional of K(t). We prove that under an anomalous rescaling Y converges in distribution to a two-dimensional Brownian motion. As a consequence, the appropriately rescaled solution of the Boltzmann equation converges to a diffusion equation

Mayakovsky’s Bedbug: Revolution, Time and Utopia

The article was submitted on 25.04.2017.In Russia, the very idea of a Communist revolution – from 1905 onwards – meant both hope and dread. This attitude is quite clearly shown in a very significant part of the Russian literary process, from 1908 to the beginning of the Stalin era. An obvious thread, in fact, connects Aleksandr Bogdanov (Red Star, 1908), Evgeny Zamyatin (We, 1921) and Vladimir Mayakovsky (The Bedbug, 1929): the growing awareness that the Communist revolution, as Lenin had conceived it, was little more than a model and that a model could not describe – much less forecast – a complex reality (a complex system) like a social and political one. As a result of this awareness, hope and a dark prophecy (Bogdanov) slowly turn into despair (Mayakovsky). The model is subsumed by Vladimir Mayakovsky’s dystopian satire of The Bedbug and The Bathhouse which propose a new paradigm of dystopia: a bottleneck in the flow of the information produced by blind adherence to a preconceived project that prevents the discovery and the implementation of la volonté générale in so complex a system as human society.Для периода господства революционных идей в России начала XX в. были характерны противоречивые настроения надежды и страха. Это ярко проявлялось и во многих произведениях русской литературы, начиная с 1908 г. и вплоть до сталинской эпохи. Такие представления были связующей нитью для творчества Александра Богданова (Красная Звезда, 1908), Евгения Замятина (Мы, 1921) и Владимира Маяковского (Клоп, 1929): по их изменениям можно проследить то, как в сознании людей росло убеждение, что коммунистическая революция – всего лишь абстрактная модель. А модель не может описать и, тем более, предсказать сложную реальность, включающую в себя социальную и политическую системы. Из осознания этого факта, по мнению автора, и происходит мрачное пророчество А. Богданова (соединенное с надеждой), которое затем перерастает в отчаяние у В. Маяковского. Эта модель представлена в сатире Маяковского – в «Клопе» и «Бане», в которых возникает новая парадигма антиутопии: информационная ограниченность, вызванная слепым следованием заранее заданному замыслу, препятствует открытию и внедрению volonté générale (всеобщей воли как результата ограничения людьми своих прав) в такой сложной системе как человеческое общество

Equivalence of QCD in the epsilon-regime and chiral Random Matrix Theory with or without chemical potential

We prove that QCD in the epsilon-regime of chiral Perturbation Theory is equivalent to chiral Random Matrix Theory for zero and both non-zero real and imaginary chemical potential mu. To this aim we prove a theorem that relates integrals over fermionic and bosonic variables to super-Hermitian or super-Unitary groups also called superbosonization. Our findings extend previous results for the equivalence of the partition functions, spectral densities and the quenched two-point densities. We can show that all k-point density correlation functions agree in both theories for an arbitrary number of quark flavors, for either mu=0 or mu=/=0 taking real or imaginary values. This implies the equivalence for all individual k-th eigenvalue distributions which are particularly useful to determine low energy constants from Lattice QCD with chiral fermions

Asymptotics of the solutions of the stochastic lattice wave equation

We consider the long time limit theorems for the solutions of a discrete wave equation with a weak stochastic forcing. The multiplicative noise conserves the energy and the momentum. We obtain a time-inhomogeneous Ornstein-Uhlenbeck equation for the limit wave function that holds both for square integrable and statistically homogeneous initial data. The limit is understood in the point-wise sense in the former case, and in the weak sense in the latter. On the other hand, the weak limit for square integrable initial data is deterministic

Long time, large scale limit of the Wigner transform for a system of linear oscillators in one dimension

We consider the long time, large scale behavior of the Wigner transform W_\eps(t,x,k) of the wave function corresponding to a discrete wave equation on a 1-d integer lattice, with a weak multiplicative noise. This model has been introduced in Basile, Bernardin, and Olla to describe a system of interacting linear oscillators with a weak noise that conserves locally the kinetic energy and the momentum. The kinetic limit for the Wigner transform has been shown in Basile, Olla, and Spohn. In the present paper we prove that in the unpinned case there exists $\gamma_0>0$ such that for any $\gamma\in(0,\gamma_0]$ the weak limit of W_\eps(t/\eps^{3/2\gamma},x/\eps^{\gamma},k), as \eps\ll1, satisfies a one dimensional fractional heat equation $\partial_t W(t,x)=-\hat c(-\partial_x^2)^{3/4}W(t,x)$ with $\hat c>0$. In the pinned case an analogous result can be claimed for W_\eps(t/\eps^{2\gamma},x/\eps^{\gamma},k) but the limit satisfies then the usual heat equation

Thermal conductivity in harmonic lattices with random collisions

We review recent rigorous mathematical results about the macroscopic behaviour of harmonic chains with the dynamics perturbed by a random exchange of velocities between nearest neighbor particles. The random exchange models the effects of nonlinearities of anharmonic chains and the resulting dynamics have similar macroscopic behaviour. In particular there is a superdiffusion of energy for unpinned acoustic chains. The corresponding evolution of the temperature profile is governed by a fractional heat equation. In non-acoustic chains we have normal diffusivity, even if momentum is conserved.Comment: Review paper, to appear in the Springer Lecture Notes in Physics volume "Thermal transport in low dimensions: from statistical physics to nanoscale heat transfer" (S. Lepri ed.