Diffusion in phase space

So far we have been considering the diffusion of concentration. The same argument will hold for the probability distribution function P(jc, t) that a particular particle is found at point x at time t since the distinction between c x, t) and V(jc, t) is, for non-interacting particles, only in the fact that W is normalized. Tims the evolution equation for the probability P(jc, 0 is written as 

Having seen the basic principle, it is easy to derive the Smoluchowski equation for a system which has many degrees of freedom. Let X, X2, , xn = jc be the set of dynamical variables describing the state of Brownian particles. To construct the Smoluchowski equation, we have to know first the relation between the average velocity v and the force F = —dUldx . Such a relation is generally written as 

The coefficients L m are called the mobility matrix, and may be obtained using hydrodynamics. It can be proved that L , is a symmetric positive definite matrix  

Given the mobility matrix, the Smoluchowski equation is obtained from the continuity equation 

The Smoluchowski equation may be regarded as a phenomenological tool for describing the fluctuation of physical quantities, and can be applied to more general situations for example the equation can be used to describe the fluctuation of thermodynamic variables (such as concentration). In such cases, the potential U(x) must be regarded as the free energy which determines the equilibrium distribution of those variables, and the relation (3.17) must be replaced by phenomenological kinetic equations. We shall see such applications in Chapter 5. 

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It is interesting to highlight that since the numerical solution of the drift term is problematic, especially when phase space is discretized, sometimes this perspective is inverted. In recent works, in fact, some continuous processes are described as if they were actually discontinuous processes. As we will see, this strategy solves some of the issues (i.e. numerical diffusion in phase space) but typically makes the problem very stiff, due to the different time scales governing the process. Readers interested in the details are referred to the work of Kumar etal. (2008b). 

Diffusion in phase space is mainly associated with the velocity, so for this term we will integrate over the interval (-oo, +oo). In the case of pure diffusion the source term of Eq. (7.75) can be calculated by using integration by parts twice ... 

Similar convection-diffusion equations to the Navier-Stokes equation can be formulated for enthalpy or species concentration. In all of these formulations there is always a superposition of diffusive and convective transport of a field quantity, supplemented by source terms describing creation or destruction of the transported quantity. There are two fundamental assumptions on which the Navier-Stokes and other convection-diffusion equations are based. The first and most fundamental is the continuum hypothesis it is assumed that the fluid can be described by a scalar or vector field, such as density or velocity. In fact, the field quantities have to be regarded as local averages over a large number of particles contained in a volume element embracing the point of interest. The second hypothesis relates to the local statistical distribution of the particles in phase space the standard convection-diffusion equations rely on the assumption of local thermal equilibrium. For gas flow, this means that a Maxwell-Boltzmann distribution is assumed for the velocity of the particles in the frame-of-reference co-moving with the fluid. Especially the second assumption may break dovm when gas flow at high temperature or low pressure in micro channels is considered, as will be discussed below. 

The Fokker-Planck equation is essentially a diffusion equation in phase space. Sano and Mozumder (SM) s model is phenomenological in the sense that they identify the energy-loss mechanism of the subvibrational electron with that of the quasi-free electron slightly heated by the external field, without delineating the physical cause of either. Here, we will briefly describe the physical aspects of this model. The reader is referred to the original article for mathematical and other details. SM start with the Fokker-Planck equation for the probability density W of the electron in the phase space written as follows ... 

The computational efficiency of a FF approach also enables simulations of dynamical behavior—molecular dynamics (MD). In MD, the classical equations of motion for a system of N atoms are solved to generate a search in phase space, or trajectory, under specified thermodynamic conditions (e.g., constant temperature or constant pressure). The trajectory provides configurational and momentum information for each atom from which thermodynamic properties such as the free energy, or time-dependent properties such as diffusion coefficients, can be calculated. 

A one-dimensional Fokker-Planck equation was used by Smoluchowski [19], and the bivariate Fokker-Planck equation in phase space was investigated by Klein [21] and Kramers [22], Note that, in essence, the Rayleigh equation [23] is a monovariate Fokker-Planck equation in velocity space. Physically, the Fokker-Planck equation describes the temporal change of the pdf of a particle subjected to diffusive motion and an external drift, manifest in the second- and first-order spatial derivatives, respectively. Mathematically, it is a linear second-order parabolic partial differential equation, and it is also referred to as a forward Kolmogorov equation. The most comprehensive reference for Fokker-Planck equations is probably Risken s monograph [14]. 

It describes the diffusion of a point in phase space for a one-dimensional Brownian particle and is used in Kramers theory. 

We will discuss this state in relation to the recent approaches of the anomalous diffusion theory [31]. It is well known [226-230] that by virtue of the divergent form of Poisson brackets (95) the evolution of the distribution function pip,q t) can be regarded as the flow of a fluid in phase space. Thus the Liouville equation (93) is analogous to the continuity equation for a fluid... 

In the rest of the discussion, we shall focus on the behavior of our model at 490 K, unless stated otherwise. When diffusion is absent, the system behaves as an excitable medium. When diffusion is included, the system behaves as an excitable or an oscillatory medium, depending on the relative gas phase pressures. In an excitable medium, the system is in a stable state and will return to that state when perturbations are applied. Upon small perturbations, the system returns to its stable state, whereby it makes only a small excursion in phase space. Often, it will turn directly back to the stable state. When the perturbation has a sufficiently large amplitude, the system will show a strong dynamic response. It will make a large excur-... 

It is natural to conceive that this short-time behavior should be due to some time interval for a trajectory to spend to look for exit ways to the next basins in the complicated structure of phase space. In the next section, we will propose a geometrical view that shows what this complexity is. Hence we consider that the hole of Na- b(t) in the short-time region should be a reflection of chaos, which is just opposite to the behavior arising from nonchaotic direct paths as observed in Hj" dynamics. The present effect is therefore expected to be more significant as the molecular size increases or the potential surface and corresponding phase-space structure become more complicated. Another important aspect of the hole in Na-,b t) is an induction time for a transport of the flow of trajectories in phase space. It is of no doubt that the RRKM theory does not take account of a finite speed for the transport of nonequilibrium phase flow from the mid-area of a basin to the transition states. Berblinger and Schlier [28] removed the contribution from the direct paths and equate the statistical part only to the RRKM rate. One should be able to do the same procedure to factor out the effect of the induction time due to transport. We believe that the transport in phase space is essentially important in a nonequilibrium rate theory and have reported a diffusion model to treat them [29]. 

One explanation for anomalous diffusion in Hamiltonian dynamics is the presence of self-similar invariant sets or hierarchical structures formed in phase space that play the role of partial barriers. They slow down the normal diffusion. A different explanation for intermittent behavior is given by the existence of deformed and approximate adiabatic invariants in phase space. They are shown in terms of elaborated perturbation theories such as the KAM and Nekhoroshev theorems. 

This concept which is based on a random walk with a well-defined characteristic waiting time (thus called a discrete-time random walk) and which applies when collisions are frequent but weak leads to the Smoluchowski equation for the evolution of the concentration of Brownian particles in configuration space. If inertial effects are included (see Note 8 of Ref. 2, due to Fiirth), we obtain the Klein-Kramers equation for the evolution of the distribution function in phase space which describes normal diffusion. The random walk considered by Einstein [2] is a walk in which the elementary steps are taken at uniform intervals in time and so is called a discrete time random walk. The concept of collisions which are frequent but weak can be clarified by remarking that in the discrete time random walk, the problem [5] is always to find the probability that the system will be in a state m at some time t given that it was in a state n at some earlier time. 

The procedure based on a generalized Chapman-Kolmogorov equation in phase space proposed by Metzler and Klafter [7,85,86] then leads, assuming the diffusion limit, to the following generalization of the Klein-Kramers... 

We remark that all the results of this section are obtained by using the B ark ai-Si I bey [30] fractional form of the Klein-Kramers equation for the evolution of the probability distribution function in phase space. In that equation, the fractional derivative, or memory term, acts only on the right-hand side—that is, on the diffusion or dissipative term. Thus, the form of the Liouville operator, or convective derivative, is preserved [cf. the right-hand side... 

This concept, which is based on a random walk with a well-defined characteristic time and which applies when collisions are frequent but weak [13], leads to the Smoluchowski equation for the evolution of the concentration of Brownian particles in configuration space. If inertial effects are included (see Note 8 of Ref. 71, due to Fiirth), we obtain the Fokker Planck equation for the evolution of the distribution function in phase space which describes normal diffusion. 

Having illustrated the problem associated with multiplicative noise we will now illustrate how the procedure is applied to obtain the drift and diffusion coefficients for the two-dimensional Fokker-Planck equation in phase space for a free Brownian particle and for the Brownian motion in a one-dimensional potential. This equation is often called the Kramers equation or Klein-Kramers equation [31]. 

It is not difficult to show that this function ip(R) satisfies the diffusion equation in Eq. (3.8). This establishes then a connection between the method of Kramers (an equivalent fictitious equilibrium problem in phase space) and the method of Kirkwood used here (a nonequilibrium problem in configuration space). 

The rate of reaction is given by the diffusion current over the potential barrier, and the energy distribution of the reacting species along the reaction coordinate is given through the density distribution in momentum space. The calculation rests, as remarked by Kramers, on the construction and solution of the equation of diffusion obeyed by a density distribution of particles in phase space. A very clear presentation of the Kramers diffusion equation has been given by Chandrasekhar.1,2... 

Nevertheless many information of several types can be obtained. The perturbations of the outer body have well defined limitations, sometimes very narrow. The three main types of motion, fly-by, bounded and oscillating of the second type - i.e. with infinitely many close approaches of the binary, approaches as close as desired, - these three main types are very interconnected in phase space. If the conjecure of Arnold diffusion... 

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