The physical implications of two forms of stochastic calculi (original) (raw)
Gaussian stochastic processes in physics
PHYSICS REPORTS (Review Section of Physics …, 1978
See section 1.9 for details. RE. Fox, Gaussian stochastic processes in physics 185 distribution, as well as averaging stochastically with respect to fr(t), the notation {. .~} will be used for u(0) averages. It should be obvious that (u(t)) = exp[_~t] u(0) and {(u(t))} = 0 (1.1.6) as can be seen using (1.1.3), (1.1.4), and (1.1.5). The average kinetic energy is M{(u(t)~u(t))} =~M exp[_2~t] {u(0) u(0)} ds ds' exp[-frts + ts')] (fr(s). P(s')) (1.1.7) =~M exp[_2~t] (3 + 3~fdsJds'exp[_~(2t-ss')] ô(s-=~kBT exp[_2Mt]+~-(iexp[_2Mt]). As t-+ m, the Brownian particle comes to thermal equilibrium with the fluid in which it is immersed. Consequently, its average kinetic energy should be~kBT.Equation (1.1.7) agrees with this if and only if AkBTa (1.1.8) which is the prototype of what is known as the fluctuation-dissipation relation [6].Once this relation is used in (1.1.7), it is seen that for all Ĩ 1W'{(u(t). u(t))} =~k 8T (1.1.9) which exhibits one aspect of the stationarity of the process. Because F(t) is assumed to be Gaussian, and (1.1.1) is a linear equation, u(t) inherits Gaussianness as a property. Following Wang and Uhlenbeck [3], this property can be used to calculate all of the statistical properties of the process in terms of a single two-time autocorrelation matrix, X11(t2defined by x~,(t2-t1) {(uI(t2)uJ(tl))} (1.1.10) = exp [-frt2 + ti)]{ui(0) u1(0)}+ f ds J ds' exp[_~(t2-s + t1-s')] 8(ss') =~Jiexp[_~(t2 + Ii)] + 2~'a exp[_~(t 2 + ti)] J ds f ds' exp[frs + SI)] 3(s-=~3iexp[-~j(t2-ti)]ô1~for t~~ti. Another aspect of stationarity is exhibited here because the time dependence involves only the difference I2-t1. In general, the autocorrelation matrix and the distribution function for a Gaussian 186 RE. Fox, Gaussian stochastic processes in physics process are related by (a~a 1) = C13 and (1.1.11) W(a1 ... a~)=(~~J!)" 2exp[_~atCya~] in which 11C'II denotes the determinant of the inverse of the correlation matrix, C 0, and the repeated indices in the exponential are to be summed from 1 to n. Therefore, for Brownian motion, the two time correlation matrix is just ({(u1(t~)u,(t5))} {(u1(t1) u,(t2))}'\-(xij(O) x~~(t2-t1) 1112 \{(u1(t~)u1(t1))} {(u1(t2) u1(t2))})-kX11(t 2-t1) Xij(0) (.. which is a 6 x 6 matrix. In spite of the large dimensionality, the ô~,in (1.1.10) means that this 6 x 6 matrix is a matrix direct product of a 2 x 2 matrix and ö,~, which is 3 x 3. Consequently, both its inverse and its determinant are easily computed. The inverse is-S'V2(u1, t1 u2, t2)~I 1 6 P2(ui, t1 u2, t2) =~. vv1i~U1, rw hich is desired. From (1.1.6) and (1.1.9) and the Gaussianness of u(t), it follows that W1(u1, t~)= (2rrkBT/M) 312 exp[_ 1~] (1.1.17) R.E Fox, Gaussian stochastic processes in physics 187 which is just the Maxwell distribution (1.1.5) and manifests the stationarity already alluded to. Equations (1.1.15), (1.1.16) and (1.1.17) give-'-3/2 / kBTI i a P 2(u1, t~u2, I2) =~~2 1r_M-~~l-exp~-2M(t2-t 1) 1~i~i\_i(u2_ uẽxp[-(a/M)(t2-t1)])~(U2~uẽxp[-(a/M)(t2-I1)]) eXP[ 2". M I 1-exp[-2(afM)(t2-t1)] Note that as (t2-t1)-*~, the initial conditioning at t1 is forgotten and P2 goes over into the Maxwell distribution for u2. Using the autocorrelation matrix, Xii' it is possible to construct three time correlation matrices, such as / xii(O) XiI(t2 t1) Xti(t3-t~) X,1(t2-t~) xii(O) xii(t3-t2) \xtit3-t1 Xii(1312) Xii(O) and in general n-time correlation matrices are constructed in an analogous fashion. Computation of their inverses and determinants leads to the higher order, n-time distribution functions [3]. If this is done for the 3 time case, it follows that the corresponding conditioned distribution, P 3(u1, t1 u2, 12; u3, t3), defined by [3]-~4 7 3(u1, I1 U2~12, U3, I3) P3(ui, t1 U2~t2, U3~I3) =~, .
Multiplicative stochastic processes in nonlinear systems. II. Canonical and noncanonical effects
Physical review, 1985
We study a physical system consisting of a low-frequency nonlinear oscillator interacting both with a thermal bath at the temperature Tl and a high-frequency linear oscillator which, in turn, interacts with a thermal bath at the temperature T2 (T2 & Tl). The interaction between slow and fast oscillator is nonlinear, thereby influencing the motion of the slow oscillator via fluctuations of a multiplicative nature. By means of a suitable procedure of elimination of the fast variables, a contracted description is obtained, which, at Tl-T2, exhibits precisely the same structure as that recently derived by Lindenberg and Seshadri [Physica (Utrecht) 109A, 483 (1981)] from the Zwanzig Hamiltonian. Instead of the transition from the overdamped to the inertial case revealed in our earlier paper [S. Faetti et a/. , Phys. Rev. A 30, 3252 (1984)] in this series, it is shown that precisely the reverse effect takes place. This is confirmed by computer calculations and the reliability of the computer calculation, in turn, is confirmed in the inertial regime via analog simulation. The theory enabling us to explore the noise-induced transition to the overdamped regime is based on an improvement of the techniques of elimination of fast variables, which produces automatic resummation over infinite perturbation terms. At T2~Tl, a space-dependent diffusion term with the same structure as that involved with the multiplicative Auctuation of the "external" kind is proven to be added to the canonical multiplicative diffusion term exhibited by the case Tl-T2. Canonical and noncanonical effects, which have so far been the subject of separate investigations, may thus be described via one single picture. It is also shown that, in the purely canonical case, the ubiquitous character of the noncanonical diffusional form, ranging from Ito-like to Stratonovich-like structure, is lost and a unique form of diffusional equation occurs. I. INTRODUCTION This paper has to be regarded as the natural continuation of that of Ref. 1 with the same research strategy based on the joint use of theory, computer calculation, and analog simulation. In Ref. 1, henceforth referred to as I, we mainly addressed the following problems. (i) First of all, we argued that it is dangerous to study a one-dimensional multiplicative differential stochastic equation without supplementing this investigation with additional information coming from a microscopic physical system behind the coarse-grained description provided by such an equation. This is indeed the general program
Physical applications of multiplicative stochastic processes
1973
The theory of multiplicative stochastic processes has been shown to lead to a density matrix description of nonequilibrium quantum mechanical phenomena. In the present paper a detailed treatment of the approach to the uniform. microcanonical, and canonical equilibrium density matrices is presented. The canonical equilibrium density matrix is approached by the density matrix which represents a subsystem in contact with a constant temperature heat reservoir. '" ",z dt Ca(t) = ~Maa,Ca,(t) + ~Maa,(t)Ca,(t), where M ,(t) = M:,Ci (t), and the following properties hold for the averaged moments of Maa' (t)l:
Contributions to the theory of multiplicative stochastic processes
Journal of Mathematical Physics, 1972
The theory of multiplicative stochastic processes is contrasted with the theory of additive stochastic processes. The case of multiplicative factors which are purely random, Gaussian, stochastic processes is treated in detail. In a spirit originally introduced by theoretical work in nuclear magnetic resonance and greatly extended by Kubo, dissipative behavior is demonstrated, on the average, for dynamical equations which do not show dissipative behavior without averaging. It is suggested that multiplicative stochastic processes lead to a conceptual foundation for nonequilibrium thermodynamics and nonequilibrium statistical mechanics, of marked generality.
Stochastic phenomena in physics
Acta Applicandae Mathematicae, 1983
The basic concepts of stochastic variables and their characterization by stochastic differential equations, diffusion equations and path integrals are reviewed. Applications of stochastic processes are then outlined for problems in optics, spin diffusion, random potentials in solids, and quantum mechanics.
Birkhäuser eBooks, 2003
Trends in Mathematics is a book series devoted to focused collections of articles arising from conferences, workshops or series of lectures. Topics in a volume may concentrate on a particular area of mathematics, or may encompass a broad range of related subject matter. The purpose of this series is both progressive and archival, a context in which to make current developments available rapidly to the community as well as to embed them in a recognizable and accessible way. Volumes of TIMS must be of high scientific qUality. Articles without proofs, or which do not contain significantly new results, are not appropriate. High quality survey papers, however, are welcome. Contributions must be submitted to peer review in a process that emulates the best journal procedures, and must be edited for correct use of language. As a rule, the language will be English, but selective exceptions may be made. Articles should conform to the highest standards of bibliographic reference and attribution. The organizers or editors of each volume are expected to deliver manuscripts in a form that is essentially "ready for reproduction." It is preferable that papers be submitted in one of the various forms of TEX in order to achieve a uniform and readable appearance. Ideally, volumes should not exceed 350-400 pages in length.
2020
The XI international conference Stochastic and Analytic Methods in Mathematical Physics was held in Yerevan 2 – 7 September 2019 and was dedicated to the memory of the great mathematician Robert Adol’fovich Minlos, who passed away in January 2018. The present volume collects a large majority of the contributions presented at the conference on the following domains of contemporary interest: classical and quantum statistical physics, mathematical methods in quantum mechanics, stochastic analysis, applications of point processes in statistical mechanics. The authors are specialists from Armenia, Czech Republic, Denmark, France, Germany, Italy, Japan, Lithuania, Russia, UK and Uzbekistan. A particular aim of this volume is to offer young scientists basic material in order to inspire their future research in the wide fields presented here.