Kinetic hierarchy and propagation of chaos in biological swarm models

We consider two models of biological swarm behavior. In these models, pairs of particles interact to adjust their velocities one to each other. In the first process, called 'BDG', they join their average velocity up to some noise. In the second process, called 'CL', one of the two particles tries to join the other one's velocity. This paper establishes the master equations and BBGKY hierarchies of these two processes. It investigates the infinite particle limit of the hierarchies at large time-scale. It shows that the resulting kinetic hierarchy for the CL process does not satisfy propagation of chaos. Numerical simulations indicate that the BDG process has similar behavior to the CL process

Celebrating Cercignani's conjecture for the Boltzmann equation

Cercignani's conjecture assumes a linear inequality between the entropy and entropy production functionals for Boltzmann's nonlinear integral operator in rarefied gas dynamics. Related to the field of logarithmic Sobolev inequalities and spectral gap inequalities, this issue has been at the core of the renewal of the mathematical theory of convergence to thermodynamical equilibrium for rarefied gases over the past decade. In this review paper, we survey the various positive and negative results which were obtained since the conjecture was proposed in the 1980s.Comment: This paper is dedicated to the memory of the late Carlo Cercignani, powerful mind and great scientist, one of the founders of the modern theory of the Boltzmann equation. 24 pages. V2: correction of some typos and one ref. adde

Complete characterization of convergence to equilibrium for an inelastic Kac model

Pulvirenti and Toscani introduced an equation which extends the Kac caricature of a Maxwellian gas to inelastic particles. We show that the probability distribution, solution of the relative Cauchy problem, converges weakly to a probability distribution if and only if the symmetrized initial distribution belongs to the standard domain of attraction of a symmetric stable law, whose index $\alpha$ is determined by the so-called degree of inelasticity, $p>0$, of the particles: $\alpha=\frac{2}{1+p}$. This result is then used: (1) To state that the class of all stationary solutions coincides with that of all symmetric stable laws with index $\alpha$. (2) To determine the solution of a well-known stochastic functional equation in the absence of extra-conditions usually adopted

Phase transitions towards frequency entrainment in large oscillator lattices

We investigate phase transitions towards frequency entrainment in large, locally coupled networks of limit cycle oscillators. Specifically, we simulate two-dimensional lattices of pulse-coupled oscillators with random natural frequencies, resembling pacemaker cells in the heart. As coupling increases, the system seems to undergo two phasetransitions in the thermodynamic limit. At the first, the largest cluster of frequency entrained oscillators becomes macroscopic. At the second, global entrainment settles. Between the two transitions, the system has features indicating self-organized criticality.Comment: 4 pages, 5 figures, submitted to PR

Raised Intracellular Calcium Contributes to Ischemia-Induced Depression of Evoked Synaptic Transmission

Oxygen-glucose deprivation (OGD) leads to depression of evoked synaptic transmission, for which the mechanisms remain unclear. We hypothesized that increased presynaptic [Ca2+]i during transient OGD contributes to the depression of evoked field excitatory postsynaptic potentials (fEPSPs). Additionally, we hypothesized that increased buffering of intracellular calcium would shorten electrophysiological recovery after transient ischemia. Mouse hippocampal slices were exposed to 2 to 8 min of OGD. fEPSPs evoked by Schaffer collateral stimulation were recorded in the stratum radiatum, and whole cell current or voltage clamp recordings were performed in CA1 neurons. Transient ischemia led to increased presynaptic [Ca2+]i, (shown by calcium imaging), increased spontaneous miniature EPSP/Cs, and depressed evoked fEPSPs, partially mediated by adenosine. Buffering of intracellular Ca2+ during OGD by membrane-permeant chelators (BAPTA-AM or EGTA-AM) partially prevented fEPSP depression and promoted faster electrophysiological recovery when the OGD challenge was stopped. The blocker of BK channels, charybdotoxin (ChTX), also prevented fEPSP depression, but did not accelerate post-ischemic recovery. These results suggest that OGD leads to elevated presynaptic [Ca2+]i, which reduces evoked transmitter release; this effect can be reversed by increased intracellular Ca2+ buffering which also speeds recovery

Optimal Hypercontractivity for Fermi Fields and Related Non-Commutative Integration

Optimal hypercontractivity bounds for the fermion oscillator semigroup are obtained. These are the fermion analogs of the optimal hypercontractivity bounds for the boson oscillator semigroup obtained by Nelson. In the process, several results of independent interest in the theory of non-commutative integration are established. {}.Comment: 18 p., princeton/ecel/7-12-9

Comment on "Why quantum mechanics cannot be formulated as a Markov process"

In the paper with the above title, D. T. Gillespie [Phys. Rev. A 49, 1607, (1994)] claims that the theory of Markov stochastic processes cannot provide an adequate mathematical framework for quantum mechanics. In conjunction with the specific quantum dynamics considered there, we give a general analysis of the associated dichotomic jump processes. If we assume that Gillespie's "measurement probabilities" \it are \rm the transition probabilities of a stochastic process, then the process must have an invariant (time independent) probability measure. Alternatively, if we demand the probability measure of the process to follow the quantally implemented (via the Born statistical postulate) evolution, then we arrive at the jump process which \it can \rm be interpreted as a Markov process if restricted to a suitable duration time. However, there is no corresponding Markov process consistent with the $Z_2$ event space assumption, if we require its existence for all times $t\in R_+$.Comment: Latex file, resubm. to Phys. Rev.