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Register a new userOn relation between generalized diffusion equations and subordination schemes
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Generalized (non-Markovian) diffusion equations with different memory kernels and subordination schemes based on random time change in the Brownian diffusion process are popular mathematical tools for description of a variety of non-Fickian diffusion processes in physics, biology and earth sciences. Some of such processes (notably, the fluid limits of continuous time random walks) allow for either kind of description, but other ones do not. In the present work we discuss the conditions under which a generalized diffusion equation does correspond to a subordination scheme, and the conditions under which a subordination scheme does possess the corresponding generalized diffusion equation. Moreover, we discuss examples of random processes for which only one, or both kinds of description are applicable.
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For reaction-diffusion processes with at most bimolecular reactants, we derive well-behaved, numerically tractable, exact Langevin equations that govern a stochastic variable related to the response field in field theory. Using duality relations, we show how the particle number and other quantities of interest can be computed. Our work clarifies long-standing conceptual issues encountered in field-theoretical approaches and paves the way for systematic numerical and theoretical analyses of reaction-diffusion problems.
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We investigate the nature of the effective dynamics and statistical forces obtained after integrating out nonequilibrium degrees of freedom. To be explicit, we consider the Rouse model for the conformational dynamics of an ideal polymer chain subject to steady driving. We compute the effective dynamics for one of the many monomers by integrating out the rest of the chain. The result is a generalized Langevin dynamics for which we give the memory and noise kernels and the effective force, and we discuss the inherited nonequilibrium aspects.
The shear stress relaxation modulus $G(t)$ may be determined from the shear stress $tau(t)$ after switching on a tiny step strain $gamma$ or by inverse Fourier transformation of the storage modulus $G^{prime}(omega)$ or the loss modulus $G^{primeprime}(omega)$ obtained in a standard oscillatory shear experiment at angular frequency $omega$. It is widely assumed that $G(t)$ is equivalent in general to the equilibrium stress autocorrelation function $C(t) = beta V langle delta tau(t) delta tau(0)rangle$ which may be readily computed in computer simulations ($beta$ being the inverse temperature and $V$ the volume). Focusing on isotropic solids formed by permanent spring networks we show theoretically by means of the fluctuation-dissipation theorem and computationally by molecular dynamics simulation that in general $G(t) = G_{eq} + C(t)$ for $t > 0$ with $G_{eq}$ being the static equilibrium shear modulus. A similar relation holds for $G^{prime}(omega)$. $G(t)$ and $C(t)$ must thus become different for a solid body and it is impossible to obtain $G_{eq}$ directly from $C(t)$.
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