Constrained Brownian motion

Constrained Brownian motion may be represented as a continuous Markov process of the / soft variables <7, ...,. A standard result of the theory of... 

The content of diffusion equation (2.175) for such a model is, moreover, independent of our choice of a system of 3,N coordinates for the unconstrained space. Constrained Brownian motion may thus be described by a model with a mobility and an effective potential /eff in any system of 3N coordinates for... 

Throughout this section, we will use the notation X (t),..., X t) to denote a unspecified set of L Markov diffusion processes when discussing mathematical properties that are unrelated to the physics of constrained Brownian motion, or that are not specific to a particular set of variables. The variables refer specifically to soft coordinates, generalized coordinates for a system of N point particles, and Cartesian particle positions, respectively. The generic variables X, ..., X will be indexed by integer variables a, p,... = 1,...,L. 

In what follows, we distinguish between the drift velocity V associated with a random variable A , which is defined by Eq. (2.223), and the corresponding drift coefficient that appears explicitly in a corresponding SDE for A , which will be denoted by A (A), and which is found to be equal to V only in the case of an Ito SDE. The values of the generalized and Cartesian drift velocities required to force each type of SDE to mimic constrained Brownian motion are determined in what follows by requiring that the resulting drift velocities have the values obtained in Section VII. 

Constrained Brownian motion may be described in generalized cordinates as the solution of a set off Ito SDEs for the soft generalized coordinates, . of the form... 

Alternatively, constrained Brownian motion of a set of pointlike particles may be described as a set of N vector SDEs for the bead positions, of the form... 

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