Soft coordinates

The symbol will thus be used to indicate a soft coordinate and 2 to indicate a coordinate that could be either soft or hard. The ranges of summation for each type of index will be those indicated above, and summation over repeated indices will be implicit, unless stated otherwise. 

The desired reduced distribution /( ) for the soft coordinates alone is given by the integral... 

The resulting conditional average is implicitly a function (A)p = (A)p( ) of the soft coordinates. Here and in what follows, )5 is used to indicate a conditional average with respect to fluctuations in the state of the surrounding solvent, for fixed values of the system s internal coordinates q and momenta p. This average over solvent degrees of freedom is unnecessary in Eq. (2.79) if A = A q,p) is a quantity (such as a bead velocity) that depends only on the system s coordinates and momenta, but is necessary if A is a quantity (such as the total force on a bead) that depends explicitly on the forces exerted on the system by surrounding solvent molecules. 

We now extend the preceding analysis to the case of a stiff system, by using an extended local-equilibrium hypothesis to remove both momenta and hard coordinates from the problem, and thus obtain a diffusion equation for the distribution of soft coordinates alone. 

To construct a diffusion equation for the soft coordinates alone, we consider a reduced distribution function... 

To relate this to the Cartesian flux velocity, we note that, in the infinitely stiff limit of interest, in which excursions from the constraint surface are negligible, the basis vectors 6R /92 may be accurately approximated by their values on the constraint surface, which are functions of the soft coordinates alone, and so may taken outside of the average average ( )f. By then using Eq. (2.107) to exclude contributions arising from the hard velocities, we identify... 

We next consider the effective force balance for all >N variables, while treating the system as an unconstrained system. For simplicity, we consider the case in which the crossover from ballistic motion to diffusion occurs at a timescale much less than any characteristic relaxation time for vibrations of the hard coordinates, so that the vibrations are overdamped, but in which the vibrational relaxation times are much less than any timescale for the diffusion of the soft coordinates. In this case, we may assume local equilibration of all 3N momenta at timescales of order the vibration time. Repeating the analysis of the Section V.A, while now treating all 3N coordinates as unconstrained, yields an effective force balance... 

Here t,- = t,( ) is one of K Lagrange multiplier fields that are functions of the soft coordinates alone, which must be chosen so as to satisfy condition (2.107) for all of the hard coordinates. Equation (2.112) has the following properties ... 

These hard forces are thus functions of the soft coordinates q alone, and completely independent of the values of c, ..., c. This is because, for this local-equilibrium distribution, the rapid variation of the hard mechanical force —dV/dc along the hard directions is canceled by a compensating variation in the Brownian contribution to... 

The dynamical reciprocal basis vectors b, ..., bj( defined above are closely related to the modified reciprocal basis vectors defined in Eq. (16.3-6) of the monograph by BCAH [4] (see the table on p. 188 of BCAH for definitions), and to a set of corresponding basis vectors used by Ottinger, which he refers to by the notation 02,/0R introduced in Eq. (5.29) of his monograph, in which 2i refers in Ottinger s notation to one of the soft coordinates. The dynamical reciprocal vectors defined here are identical to another set of... 

By comparing Eq. (2.176) to Eq. (2.78) for the desired diffusion equation, we may identify the reduced equilibrium distribution vl/ ( ) at each point on the constraint surface, to within a constant of proportionality, with the value of eq(6) on the constraint surface. In this model, the behavior of Peq(G) away from the constraint surface is dynamically irrelevant, since only the values of the derivatives of lnTeq(2) with respect to the soft coordinates, evaluated infinitesimally close to the constraint surface, enter diffusion Eq. (2.175). 

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. 

A set of Stratonovich SDEs for a constrained mechanical system may be formulated either as a set off SDEs for the soft coordinates or as a corresponding set of 3N SDEs for the Cartesian bead positions. The Stratonovich SDEs for the generalized coordinates are of the form given in Eq. (2.238), with a drift coefficient... 

The first analysis of the constrained Langevin equation was given by Fixman [9], who worked primarily in generalized coordinates. In order to resolve a discrepancy between Fixman s results and those of Hinch [10], we now follow Fixman by considering a Langevin equation for the soft coordinates, ...,... 

Here, is the mobility tensor in the chosen system of coordinates, which is a constrained mobility for a constrained system and an unconstrained mobility for an unconstrained system. As discussed in Section VII, in the case of a constrained system, Eq. (2.344) may be applied either to the drift velocities for the / soft coordinates, for which is a nonsingular / x / matrix, or to the drift velocities for a set of 3N unconstrained generalized or Cartesian coordinates, for a probability distribution (X) that is dynamically constrained to the constraint surface, for which is a singular 3N x 3N matrix. The equilibrium distribution is. (X) oc for unconstrained systems and... 

In this section, we develop some useful relationships involving the determinants and inverses of projected tensors. Let S ap be the Riemannian representation of an arbitrary symmetric covariant tensor with a Cartesian representation S v We may write the Riemannian representation in block matrix form, using the indices a,b to denote blocks in which a or p mns over the soft coordinates and i,j to represent hard coordinates, as... 

Here, we review an adiabatic approximation for the statistical mechanics of a stiff quantum mechanical system, in which vibrations of the hard coordinates are first treated quantum mechanically, while treating the more slowly evolving soft coordinates and momenta for this purpose as parameters, and in which the constrained free energy obtained by summing over vibrational quantum states is then used as a potential energy in a classical treatment of the soft coordinates and momenta. 

The soft coordinates and momenta may be treated as constants for the purpose of describing the rapid vibrations of the hard coordinates. This is the adiabatic approximation. 

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