Articles written in Pramana – Journal of Physics
Volume 11 Issue 4 October 1978 pp 379-388 Solids
Linear response theory is used to express the anelastic response (creep function and generalized compliance) of a system under an applied stress, in terms of the equilibrium strain auto-correlation. These results extend an earlier analysis to cover inhomogeneous stresses and the tensor nature of the variables. For anelasticity due to point defects, we express the strain compactly in terms of the elastic dipole tensor and the probability matrix governing dipole re-orientation and migration. We verify that re-orientations contribute to the deviatoric strain alone (Snoek, Zener, etc. effects), while the dilatory part arises solely from the long-range diffusion of the defects under a stress
Volume 11 Issue 4 October 1978 pp 389-409 Solids
The formalism of the preceding paper is applied to work out the theory of the Gorsky effect, or anelastic relaxation due to the long-range diffusion of interstitials in a host lattice, for non-interacting (low-concentration) interstitials (e.g., H in Nb). It is shown how linear response theory (LRT) provides a number of advantages that simplify the solution of the problem and permit the handling of complications due to specimen geometry and stress inhomogeneity. The multiple-relaxation time creep function of Alefeld
Volume 11 Issue 5 November 1978 pp 639-659 Solids
A density matrix formalism is developed for anelastic (mechanical) relaxation in crystalline materials with point defects characterized by elastic dipoles. The time-dependent approach to equilibrium of the strain response under the action of a constant applied stress is deduced. The formalism parallels the one used in nuclear magnetic relaxation. The anelastic relaxation time is determined as a function of the parameters occurring in the defect hopping term in the Hamiltonian. This term is responsible for the dissipation of the anelastic ‘potential’ energy into the host lattice. In a lengthy concluding section, the following aspects are discussed point by point: the advantages of the formalism presented, its scope and special cases; the physical implications of the expression obtained for the relaxation time; the similarities and differences between magnetic relaxation and anelastic relaxation, etc.
Volume 12 Issue 3 March 1979 pp 275-289 Solids
Irradiation (as in a nuclear reactor) drastically affects the defect structure and its time evolution in a material, and induces new creep mechanisms in it. We present a formalism to evaluate the contribution to creep owing to such mechanisms. Beginning with the phenomenological constitutive relation for the strain appropriate to a given mechanism, we put in simple statistical considerations to derive an expression for the corresponding creep rate. This formal expression is in terms of the defect production rate and a non-equilibrium probability distribution function involving the pertinent properties of the defect type concerned. A convenient approximation scheme for practical calculations is employed, that also makes contact with standard rate theory and provides a proper interpretation for the variables occurring there. As an illustration, we evaluate the contribution to irradiation-induced creep from the orientation-dependent shrinkage of vacancy dislocation loops in an applied stress field. The circumstances inducing transient and non-transient creep are clarified and a numerical estimate is given for the latter component.
Volume 12 Issue 4 April 1979 pp 301-315 Statistical Physics
The generalised Langevin equation (GLE), originally developed in the context of Brownian motion, yields a convenient representation for the mobility (generalised susceptibility) in terms of a frequency-dependent friction (memory function). Kubo has shown how two deep consistency conditions, or fluctuation-dissipation theorems, follow from the GLE. The first relates the mobility to the velocity auto-correlation in equilibrium, as is also derivable from linear response theory. The second is a generalised Nyquist theorem, relating the memory function to the auto-correlation of the random force driving the velocity fluctuations. Certain subtle points in the proofs of these theorems have not been dealt with sufficiently carefully hitherto. We discuss the input information required to make the GLE description a complete one, and present concise, systematic proofs starting from the GLE. Care is taken to settle the points of ambiguity in the original version of these proofs. The causality condition imposed is clarified, and Felderhof’s recent criticism of Kubo’s derivation is commented upon. Finally, we demonstrate how the ‘persistence’ of equilibrium can be used to evaluate easily the equilibrium auto-correlation of the ‘driven’ variable (e.g., the velocity) from the transient solution of the corresponding stochastic equation.
Volume 12 Issue 5 May 1979 pp 481-503 Spectroscopy
The measurement in thermal equilibrium of the vacancy contribution to the residual resistivity of metals has posed certain difficulties. The recent experiment of Celasco and co-workers represents a new, powerful approach to this problem, via the measurement of the power spectrum of the voltage noise generated by resistivity fluctuations. The latter originate in vacancy number fluctuations. We develop a theory for the power spectrum, incorporating three basic features. Vacancies can be annihilated in the material and they diffuse. Grain boundaries act as sources and sinks for vacancies. Both annihilation (a form of reaction) and diffusion are noisy processes. We therefore set up and solve a reactive-diffusive stochastic equation for the instantaneous density, with appropriate
Volume 13 Issue 4 October 1979 pp 337-352 Statistical Physics
A phenomenological interpolation model for the transition operator of a stationary Markov process is shown to be equivalent to the simplest difference approximation in the master equation for the conditional density. Comparison with the formal solution of the Fokker-Planck equation yields a criterion for the choice of the correlation time in the approximate solution. The interpolation model is shown to be form-invariant under an iteration-cum-rescaling scheme. Next, we go beyond Markov processes to find the effective time-development operator (the counterpart of the conditional density) in the following very general situation: the stochastic interruption of the systematic evolution of a variable by an arbitrary stationary sequence of randomizing pulses. Continuous-time random walk theory with a distinct first-waiting-time distribution is used, along with the interpolation model for the transition operator, to obtain the solution. Convenient closed-form expressions for the ‘averaged’ time-development operator and the autocorrelation function are presented in various special cases. These include (i) no systematic evolution, but correlated pulses; (ii) systematic evolution interrupted by an uncorrelated (Poisson) sequence of pulses.
Volume 13 Issue 5 November 1979 pp 545-557 Solids
Viscoelasticity is exhibited by polymers, metals undergoing diffusion creep, etc. The strain is a linear functional of the stress, but there is no unique equilibrium relationship between them. Under a constant stress, the strain does not saturate to an equilibrium value. This divergence is the main difficulty facing a first-principles theory of viscoelasticity, in contrast to anelasticity which has already been understood as a relaxation process in terms of response theory, fluctuations and related concepts. We now present such a theory, based on the recognition that viscoelasticity occurs whenever the spontaneous fluctuations of the strain
Volume 16 Issue 2 February 1981 pp 109-130 Statistical Physics
Diffusion with interruptions (arising from localized oscillations, or traps, or mixing between jump diffusion and fluid-like diffusion, etc.) is a very general phenomenon. Its manifestations range from superionic conductance to the behaviour of hydrogen in metals. Based on a continuous-time random walk approach, we present a comprehensive two-state random walk model for the diffusion of a particle on a lattice, incorporating arbitrary holding-time distributions for both localized residence at the sites and inter-site flights, and also the correct first-waiting-time distributions. A synthesis is thus achieved of the two extremes of jump diffusion (zero flight time) and fluid-like diffusion (zero residence time). Various earlier models emerge as special cases of our theory. Among the noteworthy results obtained are: closed-form solutions (in
Volume 16 Issue 6 June 1981 pp 437-455 Statistical Physics
Continuing our study of interrupted diffusion, we consider the problem of a particle executing a random walk interspersed with localized oscillations during its halts (
Volume 17 Issue 1 July 1981 pp 55-68 Statistical Physics
We seek the conditional probability function
Volume 21 Issue 2 August 1983 pp 111-122 Statistical Physics
We present closed expressions for the characteristic function of the first passage time distribution for biased and unbiased random walks on finite chains and continuous segments with reflecting boundary conditions. Earlier results on mean first passage times for one-dimensional random walks emerge as special cases. The divergences that result as the boundary is moved out to infinity are exhibited explicitly. For a symmetric random walk on a line, the distribution is an elliptic theta function that goes over into the known Lévy distribution with exponent 1/2 as the boundary tends to ∞.
Volume 21 Issue 3 September 1983 pp 187-200 Statistical Physics
We consider an arbitrary continuous time random walk (
Volume 37 Issue 3 September 1991 pp 235-252 Research Articles
Stochastic dynamics in the presence of quenched disorder (e.g., diffusion in a random medium) is generally treated in a suitable mean-field or effective medium approximation. While numerical simulations may help determine the accuracy of such approximations in specific models, there are relatively few instances in which analytic solutions are possible, to enable a precise comparison to be made with the mean-field results. We consider in this paper a simple but general model of quenched disorder in which a system variable
Volume 38 Issue 5 May 1992 pp 491-503 Research Articles
We solve analytically the problem of a biased random walk on a finite chain of ‘sites’ (1,2,…,
Volume 40 Issue 4 April 1993 pp 259-265
A very simple way is presented of deriving the partial differential equations (the master equations) satisfied by the probability density for certain kinds of diffusion processes in one dimension, in which the driving term is a Gaussian white noise, or a dichotomic noise, or a combination of the two. The method involves the use of certain ‘formulas of differentiation’ to derive the equations obeyed by the characteristic functions of the processes concerned, and thence the corresponding master equations. The examples presented cover a substantial number of diffusion processes that occur in physical modelling, including some master equations derived recently in the literature for generalizations of persistent diffusion.
Volume 48 Issue 1 January 1997 pp 109-128 Mathematical Aspects Of Dynamical Systems
The invariant density of one-dimensional maps in the regime of fully-developed chaos with uncorrelated additive noise is considered. Boundary conditions are shown to play a significant role in determining the precise form of the invariant density, via the manner in which they handle the spill-over, caused by the noise, of orbits beyond the interval. The known case of periodic boundary conditions is briefly recapitulated. Analytic solutions for the invariant density that are possible under certain conditions are presented with applications to specific well-known maps. The case of ‘sticky’ boundaries is generalized to ‘re-injection at the nearest boundary’, and the exact functional equations determining the invariant density are derived. Interesting boundary layer effects are shown to occur, that lead to significant modifications of the invariant density corresponding to the unperturbed (noise-free) case, even when the latter is a constant — as illustrated by an application of the formalism to the noisy tent map. All our results are non-perturbative, and hold good for any noise amplitude in the interval.
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