Reversibility and Non-reversibility in Stochastic Chemical Kinetics

Mathematical problems with mean field and local type interaction related to stochastic chemical kinetics,are considered. Our main concern various definitions of reversibility, their corollaries (Boltzmann type equations, fluctuations, Onsager relations, etc.) and emergence of irreversibility

Markov Process of Muscle Motors

We study a Markov random process describing a muscle molecular motor behavior. Every motor is either bound up with a thin filament or unbound. In the bound state the motor creates a force proportional to its displacement from the neutral position. In both states the motor spend an exponential time depending on the state. The thin filament moves at its velocity proportional to average of all displacements of all motors. We assume that the time which a motor stays at the bound state does not depend on its displacement. Then one can find an exact solution of a non-linear equation appearing in the limit of infinite number of the motors.Comment: 10 page

On products of skew rotations

Let $H_1(p,q)$, $H_2(p,q)$ be two time-independent Hamiltonians with one degree of freedom and $\{S_1^t\}$, $\{S_2^t\}$ be the one-parametric groups of shifts along the orbits of Hamiltonian systems generated by $H_1$, $H_2$. In some problems of population genetics there appear the transformations of the plane having the form $T^{(h_1,h_2)}=S^{h_2}_2\cdot S_1^{h_1}$ under some conditions on $H_1$, $H_2$. We study in this paper asymptotical properties of trajectories of $T^{(h_1,h_2)}$.Comment: 13 pages, 10 figure

Rigorous Analysis of Singularities and Absence of Analytic Continuation at First Order Phase Transition Points in Lattice Spin Models

We report about two new rigorous results on the non-analytic properties of thermodynamic potentials at first order phase transition. The first one is valid for lattice models ($d\geq 2$) with arbitrary finite state space, and finite-range interactions which have two ground states. Under the only assumption that the Peierls Condition is satisfied for the ground states and that the temperature is sufficiently low, we prove that the pressure has no analytic continuation at the first order phase transition point. The second result concerns Ising spins with Kac potentials $J_\gamma(x)=\gamma^d\phi(\gamma x)$, where $0<\gamma<1$ is a small scaling parameter, and $\phi$ a fixed finite range potential. In this framework, we relate the non-analytic behaviour of the pressure at the transition point to the range of interaction, which equals $\gamma^{-1}$. Our analysis exhibits a crossover between the non-analytic behaviour of finite range models ($\gamma>0$) and analyticity in the mean field limit ($\gamma\searrow 0$). In general, the basic mechanism responsible for the appearance of a singularity blocking the analytic continuation is that arbitrarily large droplets of the other phase become stable at the transition point.Comment: 4 pages, 2 figure

A Search for Small-Scale Clumpiness in Dense Cores of Molecular Clouds

We have analyzed HCN(1-0) and CS(2-1) line profiles obtained with high signal-to-noise ratios toward distinct positions in three selected objects in order to search for small-scale structure in molecular cloud cores associated with regions of high-mass star formation. In some cases, ripples were detected in the line profiles, which could be due to the presence of a large number of unresolved small clumps in the telescope beam. The number of clumps for regions with linear scales of ~0.2-0.5 pc is determined using an analytical model and detailed calculations for a clumpy cloud model; this number varies in the range: ~2 10^4-3 10^5, depending on the source. The clump densities range from ~3 10^5-10^6 cm^{-3}, and the sizes and volume filling factors of the clumps are ~(1-3) 10^{-3} pc and ~0.03-0.12. The clumps are surrounded by inter-clump gas with densities not lower than ~(2-7) 10^4 cm^{-3}. The internal thermal energy of the gas in the model clumps is much higher than their gravitational energy. Their mean lifetimes can depend on the inter-clump collisional rates, and vary in the range ~10^4-10^5 yr. These structures are probably connected with density fluctuations due to turbulence in high-mass star-forming regions.Comment: 23 pages including 4 figures and 4 table

NGC 7538 : Multiwavelength Study of Stellar Cluster Regions associated with IRS 1-3 and IRS 9 sources

We present deep and high-resolution (FWHM ~ 0.4 arcsec) near-infrared (NIR) imaging observations of the NGC 7538 IRS 1-3 region (in JHK bands), and IRS 9 region (in HK bands) using the 8.2m Subaru telescope. The NIR analysis is complemented with GMRT low-frequency observations at 325, 610, and 1280 MHz, molecular line observations of H13CO+ (J=1-0), and archival Chandra X-ray observations. Using the 'J-H/H-K' diagram, 144 Class II and 24 Class I young stellar object (YSO) candidates are identified in the IRS 1-3 region. Further analysis using 'K/H-K' diagram yields 145 and 96 red sources in the IRS 1-3 and IRS 9 regions, respectively. A total of 27 sources are found to have X-ray counterparts. The YSO mass function (MF), constructed using a theoretical mass-luminosity relation, shows peaks at substellar (~0.08-0.18 Msolar) and intermediate (~1-1.78 Msolar) mass ranges for the IRS 1-3 region. The MF can be fitted by a power law in the low mass regime with a slope of Gamma ~ 0.54-0.75, which is much shallower than the Salpeter value of 1.35. An upper limit of 10.2 is obtained for the star to brown dwarf ratio in the IRS 1-3 region. GMRT maps show a compact HII region associated with the IRS 1-3 sources, whose spectral index of 0.87+-0.11 suggests optical thickness. This compact region is resolved into three separate peaks in higher resolution 1280 MHz map, and the 'East' sub-peak coincides with the IRS 2 source. H13CO+ (J=1-0) emission reveals peaks in both IRS 1-3 and IRS 9 regions, none of which are coincident with visible nebular emission, suggesting the presence of dense cloud nearby. The virial masses are approximately of the order of 1000 Msolar and 500 Msolar for the clumps in IRS 1-3 and IRS 9 regions, respectively.Comment: 27 pages, 18 figures, 5 tables. Accepted for publication in MNRA

Random Walks and Chemical Networks

Projet MEVALWe consider continuous time random walks in the orthant with bounded jumps, the rates however are not bounded - they have a polynomial dependence on the coordinates of the point. The case when the rates are bounded correspon- ds in applications to the queueing theory, more exactly to markovian communication networks. The goal of this paper is to discuss the situation for polynomial rates, we show that the boundaries do not play role, but new effects and complicated behaviour can arise due to different time scales