dynamic critical exponent of the bfacf algorithm for self avoiding walks we study the dynamic critical behavior of the bfacf algorithm for generating self avoiding walks with variable length and fixed endpoints we argue theoretically and confirm by monte carlo simulations in dimensions 2 3 and 4 that the autocorrelation time scales as int nr 4r n 4v

monte carlo generation of self avoiding walks with fixed endpoints and fixed length we propose a new class of dynamic monte carlo algorithms for generating self avoiding walks uniformly from the ensemble with fixed endpoints and fixed length in any dimension and prove that these algorithms are ergodic in all cases we also prove the ergodicity of a variant of the pivot algorithm

the spectral gap for some spin chains with discrete symmetry breaking we prove that for any finite set of generalized valence bond solid gvbs states of a quantum spin chain there exists a translation invariant finite range hamiltonian for which this set is the set of ground states this result implies that there are gvbs models with arbitrary broken discrete symmetries that are described as combinations of lattice translations lattice reflections and local unitary or anti unitary transformations we also show that all gvbs models that satisfy some natural conditions have a spectral gap the existence of a spectral gap is obtained by applying a simple and quite general strategy for proving lower bounds on the spectral gap of the generator of a classical or quantum spin dynamics this general scheme is interesting in its own right and threfore although the basic idea is not new we present it in a system independent setting the results are illustrated with a number of examples

current fluctuations of the one dimensional symmetric simple exclusion process with step initial condition for the symmetric simple exclusion process on an infinite line we calculate exactly the fluctuations of the integrated current qt during time t through the origin when in the initial condition the sites are occupied with density a on the negative axis and with density b on the positive axis all the cumulants of qt grow like sqrt t in the range where q t sim sqrt t the decay exp qt3 t of the distribution of qt is non gaussian our results are obtained using the bethe ansatz and several identities derived recently by tracy and widom for exclusion processes on the infinite line

einstein relation for a class of interface models a class of sos interface models which can be seen as simplified stochastic ising model interfaces is studied in the absence of an external field the long time fluctuations of the interface are shown to behave as brownian motion with diffusion coefficient given by a green kubo formula when a small external field h is applied it is shown that the shape of the interface converges exponentially fast to a stationary distribution and the interface moves with an asymptotic velocity v h the mobility is shown to exist and to satisfy the einstein relation where is the inverse temperature

remarks on scaling a model of witten sander type we try to prove rigorously that the perimeter of the large witten sander cluster does not scale as the square root of its area by making a forced comparison with the ill posed hele shaw problem of fluid dynamics the attempt is not completely successful nevertheless some interesting consequences of the comparison are derived

stretched exponential fixation in stochastic ising models at zero temperature we study a class of continuous time markov processes which describes 1 spin flip dynamics on the hypercubic lattice d d 2 with initial spin configurations chosen according to the bernoulli product measure with density p of spins 1 during the evolution the spin at each site flips at rate c 0 or 0 0 q t exp t 1 d for large t in d 2 we obtain the complementary bound for arbitrary 0 q t exp t 1 2 for large t

metastability for the exclusion process with mean field interaction we consider an exclusion particle system with long range mean field type interactions at temperature 1 the hydrodynamic limit of such a system is given by an integrodifferential equation with one conservation law on the circle c it is the gradient flux of the kac free energy functional f for 1 any constant function with value m 1 1 is the global minimizer of f in the space u int c u x dx m for 1 f restricted to u int c u x dx m may have several local minima in particular the constant solution may not be the absolute minimizer of f we therefore study the long time behavior of the particle system when the initial condition is close to a homogeneous stable state giving results on the time of exit from suitable subsets of its domain of attraction we follow the freidlin wentzell approach first we study in detail f together with the time asymptotics of the solution of the hydrodynamic equation then we study the probability of rare events for the particle system i e large deviations from the hydrodynamic limit

hitting times of rare events in markov chains a general condition for a markov process with finite states is given for the convergence of the normalized hitting time of a rare event to a mean one exponential law it is then applied to the boundary driven symmetric simple exclusion process for an open set of density profiles which does not contain the stationary density profile

absence of mass gap for a class of stochastic contour models we study a class of markovian stochastic processes in which the state space is a space of lattice contours and the elementary motions are local deformations we show under suitable hypotheses on the jump rates that the infinitesimal generator has zero mass gap this result covers among others the bfacf dynamics for fixed endpoint self avoiding walks and the sterling greensite dynamics for fixed boundary self avoiding surfaces our models also mimic the glauber dynamics for the low temperature ising model the proofs are based on two new general principles the minimum hitting time argument and the mean or mean exponential hitting time argument

stretched exponential decay in a kinetic ising model with dynamical constraint we show that for the standard nearest neighbor spin flip dynamics in one dimension with the constraint of constant energy the spin spin correlation function decays as exp c sqrt t for larget we prove an upper and lower bound the coefficientc of the lower bound is given as the solution of a variational problem and is conjectured to be exact

canonical structure and orthogonality of forces and currents in irreversible markov chains we discuss a canonical structure that provides a unifying description of dynamical large deviations for irreversible finite state markov chains continuous time onsager theory and macroscopic fluctuation theory mft for markov chains this theory involves a non linear relation between probability currents and their conjugate forces within this framework we show how the forces can be split into two components which are orthogonal to each other in a generalised sense this splitting allows a decomposition of the pathwise rate function into three terms which have physical interpretations in terms of dissipation and convergence to equilibrium similar decompositions hold for rate functions at level 2 and level 2 5 these results clarify how bounds on entropy production and fluctuation theorems emerge from the underlying dynamical rules we discuss how these results for markov chains are related to similar structures within mft which describes hydrodynamic limits of such microscopic models

unimodality convexity and applications properties of univariate unimodal distributions concepts of multivariate unimodality some more notions of unimodality unimodality for discrete distributions unimodality of infinitely divisible distributions unimodality and notions of dependence ordering of distributions by peakedness applications of unimodality in statistical inference convexity in reliability theory appendix

application of log sobolov inequality to the stochastic dynamics of unbounded spin systems on the lattice abstract we consider a ferromagnetic lattice spin system with unbounded spins and investigate the relaxation property for the associated stochastic dynamics the glauber dynamics in the finite volume case we prove that the following two conditions are equivalent 1 the log sobolev inequality for the finite volume gibbs states holds uniformly in both the volume and the boundary condition 2 the finite volume glauber dynamics relaxes to equilibrium exponentially fast uniformly in the volume whenever it starts from a tempered configuration this can be considered as a complementary result to the ones previously obtained for infinite volume glauber dynamics by b zegarlinski 1996 comm math phys 175 401 432 our result can also be viewed as an extension of the equivalence theorem known for compact spin space settings

nonlocal monte carlo algorithm for self avoiding walks with fixed endpoints we study a new monte carlo algorithm for generating self avoiding walks with variable length controlled by a fugacity and fixed endpoints the algorithm is a hybrid of local bfacf and nonlocal cut and paste moves we find that the critical slowing down measured in units of computer time is reduced compared to the pure bfacf algorithm cpu n 2 3 versus n 3 0 we also prove some rigorous bounds on the autocorrelation time for these and related monte carlo algorithms

hydrodynamics and large deviation for simple exclusion processes we prove the hydrodynamical limit for weakly asymmetric simple exclusion processes a large deviation property with respect to this limit is established for the symmetric case we treat also the situation where a slow reaction creation and annihilation of particles is present

bound on the mass gap for a stochastic contour model at low temperature a two dimensional stochastic contour model is considered in which the state space consists of simple closed lattice contours about a fixed origin and the evolution in the state space is a continuous time reversible jump process an upper bound is obtained on the mass gap for the model which goes rapidly to zero in a low temperature limit a connection between this model and statistical mechanical and droplet models is discussed

optimum partition procedures for separating good and bad treatments abstract this article considers partitioning a given set of competing treatments into two subsets with the purpose of separating good and bad treatments the qualities good and bad are defined in terms of the parameter vector of the underlying distribution family regarding the probability of an incorrect partition within the class of so called natural procedures minimax solutions are derived under suitable distributional assumptions these results extend previous investigations to the larger family of quasi concave distributions it is shown that the multivariate noncentral t distribution with v degrees of freedom is 1 v concave and hence quasi concave thus the unknown variance case in a normal distribution setup is covered as a special application specifically for partitioning with respect to a control the presented minimax results are used in the unknown variance case to determine optimal allocations of total sample sizes that guarantee a correct partition at a preassigned confidence lev

monte carlo study of self avoiding surfaces two models of self avoiding surfaces on the cubic lattice are studied by monte carlo simulations both the first model with fluctuating boundary and the second one with a fixed boundary are found to belong to the universality class of branched polymers the algorithms as well as the methods used to extract the critical exponents are described in detail the results are compared to other recent estimates in the literature

rigorous derivation of domain growth kinetics without conservation laws the time evolution of the ising model that describes shrinking domains is studied a singly connected domain of ising spins embedded in a sea of the opposite phase develops att 0 according to a dynamic rule that does not allow its perimeter to increase at long enough times the domain disappears we show that the average lifetime of such a domain is proportional to its area we also consider thet 0 dynamics of a single infinite quadrant the area of the quadrant decreases during the time evolution and we show that the area lost grows linearly with time we solve a first passage time problem as well that is we calculate the average time it takes for the area lost to reach a given value for the first time lastly we map the infinite quadrant model onto a diffusion problem with exclusion in one dimension this latter problem is mapped onto a critical six vertex model

convergence speed for simple symmetric exclusion an explicit calculation for the infinite volume simple symmetric nearest neighbor exclusion process in one dimension we investigate the speed of convergence to equilibrium from a particular initial distribution we use duality to reduce the analysis to that of the two particle process which we further reduce to a random walk reflecting rightward at zero whose generator is self adjoint on l 2 z we obtain the spectral representation of the generator and use asymptotic analysis to show that convergence is slow

bound on the mass gap for finite volume stochastic ising models at low temperature we consider a sequence of finite volume z d d 2 reversible stochastic ising models in the low temperature regime and having invariant measures satisfying free boundary conditions we show that associated with the models are random hitting times whose expectations regarded as a function of grow exponentially in d 1 d moreover the mass gaps for the models shrink exponentially fast in d 1 d a geometrical lemma is employed in the analysis which states that if a peierls contour is sufficiently small relative to the faces of then the fraction of the contour tangent to the faces is less than a constant smaller than one

nonergodicity of local length conserving monte carlo algorithms for the self avoiding walk it is proved that every dynamic monte carlo algorithm for the self avoiding walk based on a finite repertoire of local n conserving elementary moves is nonergodic heren is the number of bonds in the walk indeed for largen each ergodic class forms an exponentially small fraction of the whole space this invalidates at least in principle the use of the verdier stockmayer algorithm and its generalizations for high precision monte carlo studies of the self avoiding walk

on the relaxation processes of the one dimensional kinetic ising model the author studies various relaxation phenomena of the general single spin flip one dimensional kinetic ising model with transition rates wi sigma i 1 2 alpha 1 delta sigma i 1 sigma i 1 1 1 2 gamma sigma i sigma i 1 sigma i 1 they show that the long time behaviour can be described by an exponential decay except at some special parameter values contrary to previous numerical results furthermore he proves that the exponent z characterizing the disappearance of the excitation gap is not restricted the case of delta 1 is investigated in detail

approach to equilibrium for random walks on graphs and for stochastic infinite particle processes associated with a finite connected graph g is a doubly stochastic laplacian matrix g g a lower bound on the mass gap for g g is obtained i e the first nonzero eigenvalue of g g this estimate is then used to estimate the mass gap for some infinite particle stochastic generators with corresponding processes that are closely related to monte carlo methods employed in statistical mechanics calculations

large deviations for a catalytic fleming viot branching system we consider a jump diffusion process describing a system of diffusing particles that upon contact with an obstacle catalyst die and are replaced by an independent offspring with position chosen according to a weighted average of the remaining particles the obstacle is a bounded nonnegative function v x and the birth death mechanism is similar to the fleming viot critical branching since the mass is conserved we prove a hydrodynamic limit for the empirical measure identified as the solution to a generalized semilinear reaction diffusion equation with nonlinearity given by a quadratic operator a large deviation principle from the deterministic hydrodynamic limit is provided the upper bound is given in any dimension and the lower bound is proven for d 1 and v bounded away from 0 an explicit formula for the rate function is provided via an orlicz type space 2006 wiley periodicals inc

large deviations of interacting particle systems in infinite volume we prove a large deviation theorem from the hydrodynamical limit for the empirical measure of ginzburg landau and zero range processes in infinite volume starting from deterministic initial configurations in the ginzburg landau case the main tool is the study of the evolution of the h 1 norm and in the zero range case the attractiveness which allows couplings

on the swendsen wang dynamics i exponential convergence to equilibrium we present rigorous results on the exponential convergence to equilibrium for the swendsen wang stochastic dynamics for thed dimensional ising ferromagnet with external magnetic fieldh in the thermodynamic limit we consider various situations mainly in the low temperature regime in which boundary conditions are homogeneous and parallel or opposite to the external field in the latter case we relate directly the tunneling from the metastable phase to the stable one with the exponential convergence to equilibrium

the pivot algorithm and polygons results on the fcc lattice the authors consider the pivot algorithm for polygons on the face centred cubic lattice and prove ergodicity a numerical study of the algorithm is carried out in particular the authors look at the acceptance fraction of the elementary moves and find that it goes only slowly to zero as the length of the polygons is increased the numerical properties of polygons on the fcc lattice are also considered they calculate the exponent nu by considering the mean square radius of gyration of the polygons and compare the result to the expected value from field theory they also consider the mean span of the polygons and discuss corrections to scaling and their influence on the numerical calculation of critical exponents

join and cut algorithm for self avoiding walks with variable length and free endpoints we introduce a new monte carlo algorithm for generating self avoiding walks of variable length and free endpoints the algorithm works in the unorthodox ensemble consisting of all pairs of saws such that the total number of stepsntot in the two walks is fixed the elementary moves of the algorithm are fixed n e g pivot moves on the individual walks and a novel join and cut move that concatenates the two walks and then cuts them at a random location we analyze the dynamic critical behavior of the new algorithm using a combination of rigorous heuristic and numerical methods in two dimensions the autocorrelation time in cpu units grows as n 1 5 and the behavior improves in higher dimensions this algorithm allows high precision estimation of the critical exponent

rapid convergence to equilibrium in one dimensional stochastic ising models we consider one dimensional stochastic ising models with finite range interactions for such processes we first prove that the semi group of the process converges exponentially fast on the l space of the gibbs states under the additional hypothesis that the flip rates are attractive we prove that the semigroup acting on the cylinder functions converges to equilibrium exponentially fast in the uniform norm

nonlinear diffusion limit for a system with nearest neighbor interactions we consider a system of interacting diffusions the variables are to be thought of as charges at sites indexed by a periodic one dimensional lattice the diffusion preserves the total charge and the interaction is of nearest neighbor type with the appropriate scaling of lattice spacing and time a nonlinear diffusion equation is derived for the time evolution of the macroscopic charge density

current fluctuations in one dimensional diffusive systems with a step initial density profile we show how to apply the macroscopic fluctuation theory mft of bertini de sole gabrielli jona lasinio and landim to study the current fluctuations of diffusive systems with a step initial condition we argue that one has to distinguish between two ways of averaging the annealed and the quenched cases depending on whether we let the initial condition fluctuate or not although the initial condition is not a steady state the distribution of the current satisfies a symmetry very reminiscent of the fluctuation theorem we show how the equations of the mft can be solved in the case of non interacting particles the symmetry of these equations can be used to deduce the distribution of the current for several other models from its knowledge derrida and gerschenfeld in j stat phys 136 1 15 2009 for the symmetric simple exclusion process in the range where the integrated current q t sim sqrt t we show that the non gaussian decay exp q t 3 t of the distribution of q t is generic

phase segregation dynamics in particle systems with long range interactions i macroscopic limits we present and discuss the derivation of a nonlinear nonlocal integrodifferential equation for the macroscopic time evolution of the conserved order parameter r t of a binary alloy undergoing phase segregation our model is ad dimensional lattice gas evolving via kawasaki exchange with respect to the gibbs measure for a hamiltonian which includes both short range local and long range nonlocal interactions the nonlocal part is given by a pair potential dj x y 0 x and y in d in the limit 0 the macroscopic evolution is observed on the spatial scale 1 and time scale 2 i e the density r t is the empirical average of the occupation numbers over a small macroscopic volume element centered atr x a rigorous derivation is presented in the case in which there is no local interaction in a subsequent paper part ii we discuss the phase segregation phenomena in the model in particular we argue that the phase boundary evolutions arising as sharp interface limits of the family of equations derived in this paper are the same as the ones obtained from the corresponding limits for the cahn hilliard equation

ising models on the lattice sierpinski gasket ferromagnetic ising models on the lattice sierpinski gasket are considered we prove the dobrushin shlosmann mixing condition and discuss corresponding properties of the stochastic ising models

exact simultaneous confidence intervals for multiple comparisons among three or four mean values abstract a procedure for constructing exact simultaneous confidence intervals for a finite set of multiple comparisons among three or four groups is presented these confidence intervals are hyperbolic and based on an unbiased estimator of the mean values having a normal distribution with covariance matrix 2v where v is an arbitrary known positive definite matrix this procedure is used to study the amount of conservativeness of tukey kramer intervals for pairwise comparisons in an unbalanced one way analysis of variance design

new monte carlo method for the self avoiding walk we introduce a new monte carlo algorithm for the self avoiding walk saw and show that it is particularly efficient in the critical region long chains we also introduce new and more efficient statistical techniques we employ these methods to extract numerical estimates for the critical parameters of the saw on the square lattice we find 2 63820 0 00004 0 00030 1 352 0 006 0 025 v 0 7590 0 0062 0 0042 where the first error bar represents systematic error due to corrections to scaling subjective 95 confidence limits and the second bar represents statistical error classical 95 confidence limits these results are based on saws of average length 166 using 340 hours cpu time on a cdc cyber 170 730 we compare our results to previous work and indicate some directions for future research

metastability and exponential approach to equilibrium for low temperature stochastic ising models we present a proof of the exponential convergence to equilibrium of single spin flip stochastic dynamics for the two dimensional ising ferromagnet in the low temperature case with not too small external magnetic fieldh uniformly in the volume and in the boundary conditions

exponential convergence to equilibrium for a class of random walk models we prove exponential convergence to equilibrium l 2 geometric ergodicity for a random walk with inward drift on a sub cayley rooted tree this randomwalk model generalizes a monte carlo algorithm for the self avoiding walk proposed by berretti and sokal if the number of vertices of leveln in the tree grows asc n n n 1 we prove that the autocorrelation time satisfies n 2 n 1

a criterion for invariant measures of markov processes using the martingale formulation for markov processes introduced by stroock and varadhan we develop a criterion for checking if a measure happens to be invariant

the l2spectral gap of certain positive recurrent markov chains and jump processes let d denote the generator of a continuous time positive recurrent markov process with state space n or r sufficient conditions are given to imply the existence of 0 such that if 0 is a point of the spectrum of d considered as an operator on the l 2 space of the equilibrium distribution then re a related result is given for discrete time markov chains

uniform and l 2 convergence in one dimensional stochastic ising models we study the rate of convergence to equilibrium of one dimensional stochastic ising models with finite range interactions we donot assume that the interactions are ferromagnetic or that the flip rates are attractive the infinitesimal generators of these processes all have gaps between zero and the rest of their spectra we prove that if one of these processes is observed by means of local observables then the convergence is seen to be exponentially fast with an exponent that is any number less than the spectral gap moreover this exponential convergence is uniform in the initial configuration

kinetically constrained spin models we analyze the density and size dependence of the relaxation time for kinetically constrained spin models kcsm intensively studied in the physics literature as simple models sharing some of the features of the glass transition kcsm are interacting particle systems on mathbb z d with glauber like dynamics reversible w r t a simple product i i d bernoulli p measure the essential feature of a kcsm is that the creation destruction of a particle at a given site can occur only if the current configuration around it satisfies certain constraints which completely define each specific model no other interaction is present in the model from the mathematical point of view the basic issues concerning positivity of the spectral gap inside the ergodicity region and its scaling with the particle density p remained open for most kcsm with the notably exception of the east model in d 1 aldous and diaconis in j stat phys 107 5 6 945 975 2002 here for the first time we i identify the ergodicity region by establishing a connection with an associated bootstrap percolation model ii develop a novel multi scale approach which proves positivity of the spectral gap in the whole ergodic region iii establish sometimes optimal bounds on the behavior of the spectral gap near the boundary of the ergodicity region and iv establish pure exponential decay at equilibrium for the persistence function i e the probability that the occupation variable at the origin does not change before time t our techniques are flexible enough to allow a variety of constraints and our findings disprove certain conjectures which appeared in the physical literature on the basis of numerical simulations

proof of a conjecture for the perturbed gelfand equation from combustion theory abstract in this paper we prove that when the domain b is a two dimensional ball for all small positive e the solution curve u of the perturbed gelfand equation u eu 1 eu with u b 0 is exactly s shaped this settles a long standing conjecture on the exact number of solutions to this equation

on the universality class of planar self avoiding surfaces with fixed boundary using a modified version of a monte carlo algorithm proposed by sterling and greensite 1983 the authors obtain the exponents theta 1 51 or 0 25 and nu 0 502 or 0 024 for planar self avoiding surfaces with a fixed boundary in three dimensions consistent with the conjectured exact values for branched polymers they show how the modifications are needed to obtain a viable distribution of surfaces

nonexponential relaxation of ising models within the droplet picture we develop a unified droplet description for the low temperature phase of various ising systems in order to explain their slow relaxation phenomena for the equilibrium autocorrelation function we find log c t tn behaviour where n depends only on the type of randomness and on the dimension 0 n 1 this decay function holds even in the so called griffiths phase for intermediate times

coupling concentration inequalities and stochastic dynamics in the context of interacting particle systems we study the influence of the action of the semigroup on the concentration property of lipschitz functions as an application this gives a new approach to estimate the relaxation speed to equilibrium of interacting particle systems we illustrate our approach in a variety of examples for which we obtain several new results with short and nontechnical proofs these examples include the symmetric and asymmetric exclusion processes and high temperature spin flip dynamics glauber dynamics we also give a new proof of the poincare inequality based on coupling in the context of one dimensional gibbs measures in particular we cover the case of polynomially decaying potentials where the log sobolev inequality does not hold

multigrid methods convergence theory in a variational framework to solve the symmetric variational problem u in h a u v l v forall v in h multigrid methods are constructed using a sequence of nested subspaces u k k 0 to m of h the convergence is studied relatively to the energy norm i e in the hilbert space h a cdot cdot and is based on the 2 level case m 1 for the m 1 level case rather sharp bounds for the convergence factor can be constructed recursively and their nondependence on the number of levels is established under certain hypothesis

bounds on the spectrum for markov chains and markov processes a generalization of cheeger s inequality we prove a general version of cheeger s inequality for discrete time markov chains and continuous time markovian jump processes both reversible and nonreversible with general state space we also prove a version of cheeger s inequality for markov chains and processes with killing as an application we prove l 2 exponential convergence to equilibrium for random walk with inward drift on a class of countable rooted graphs

spatial statistics and bayesian computation on wednesday may 6th 1992 professor b w silverman in the chair summary markov chain monte carlo mcmc algorithms such as the gibbs sampler have provided a bayesian inference machine in image analysis and in other areas of spatial statistics for several years founded on the pioneering ideas of ulf grenander more recently the observation that hyperparameters can be included as part of the updating schedule and the fact that almost any multivariate distribution is equivalently a markov random field has opened the way to the use of mcmc in general bayesian computation in this paper we trace the early development of mcmc in bayesian inference review some recent computational progress in statistical physics based on the introduction of auxiliary variables and discuss its current and future relevance in bayesian applications we briefly describe a simple mcmc implementation for the bayesian analysis of agricultural field experiments with which we have some practical experience

scaling limits of weakly asymmetric interfaces we consider three models of evolving interfaces intimately related to the weakly asymmetric simple exclusion process with n particles on a finite lattice of 2n sites our model 1 defines an evolving bridge on 0 1 our model 1 w an evolving excursion on 0 1 while our model 2 consists of an evolving pair of non crossing bridges on 0 1 based on the observation that the invariant measures of the dynamics depend on the area under or between the interface s we characterise the scaling limits of the invariant measures when the asymmetry of the exclusion process scales like n frac 3 2 then we show that the scaling limits of the dynamics themselves are expressed in terms of variants of the stochastic heat equation in particular in model 1 w we obtain the well studied reflected stochastic heat equation introduced by nualart and pardoux probab theory relat fields 93 1 77 89 1992

semilinear elliptic boundary value problems in chemical reactor theory abstract this paper is devoted to the study of semilinear elliptic boundary value problems arising in chemical reactor theory which obey the simple arrhenius rate law and newtonian cooling we prove that ignition and extinction phenomena occur in the stable steady temperature profile at some critical values of a dimensionless heat evolution rate