depends on the system, the properties of interest and the required accuracy.

The choice must be judiciously made. It is highly desirable and bene¬cial

for the approximations that will be made, that the “irrelevant” degrees of

freedom equilibrate faster (and preferably much faster) than the “relevant”

degrees of freedom, as the approximations will unavoidably produce errors

on the time scale were these two overlap. However, such a clear distinction

is generally not possible, and one must accept the inaccurate prediction of

dynamic details of the “relevant” degrees of freedom on short time scales.

Some examples are listed below:

• A rather spherical molecule, as CCl4 . Relevant: the center-of-mass mo-

tion; irrelevant: the rotational and internal degrees of freedom.

• A (macro)molecule in a solvent. Relevant: all atoms of the solute; irrel-

evant: all solvent molecules. With this choice there is certainly overlap

between the time ranges for the two sets of particles. For example, for a

protein in water one may expect incorrect dynamic behavior of charged

side chains on a time scale shorter than, or comparable to, the dielectric

relaxation time of the solvent.

• A large linear polymer. Relevant: the centers of mass of groups of n con-

secutive atoms; irrelevant: all other degrees of freedom. This may work if

the polymer shows self-similar behavior, i.e., that macroscopic properties

scale in some regular manner with n. These are typical superatom models

(see Section 8.4).

• A protein (or other compact non-selfsimilar macromolecule). Relevant:

a subset of atoms (as C± atoms, or backbone atoms, or backbone atoms

plus a simpli¬ed side chain representation); irrelevant: all other atoms

or degrees of freedom including the surrounding solvent. This is also a

superatom approach.

• A protein (or other compact macromolecule). Relevant: a set of collective

“essential degrees of freedom” generated from an analysis of a detailed

simulation, e.g., the ¬rst few eigenvectors with largest eigenvalue from

a principal component analysis based on atomic ¬‚uctuations, or from a

quasi-harmonic analysis. Irrelevant: all other eigenvectors.

• A chemical reaction or other infrequent process in a complex system.

Relevant: the reaction coordinate, being a function of internal degrees of

freedom of the system that captures the important path between reactants

and products in a chemical reaction. This may concern one dimension, or

encompass a space of a few dimensions. Irrelevant: all other degrees of

freedom.

• A colloidal dispersion of relatively large spherical rigid particles in a sol-

8.2 The generalized Langevin equation 251

vent. Relevant: center of mass coordinates of the particles. Irrelevant:

rotational and internal degrees of freedom of the particles, and solvent

degrees of freedom. For non-spherical particles their rotational degrees of

freedom may be considered relevant as well.

• A rather homogeneous condensed phase under slowly varying external

in¬‚uences. Relevant: Densities of molecular components at grid points on

a chosen 3D spatial grid; irrelevant: all other degrees of freedom. Instead

of a regular 3D grid one may use other, possibly time-dependent, ¬nite

element subdivisions of space.

In some cases we can choose cartesian degrees of freedom as the relevant

ones (e.g., when we divide the particles over both classes), but in most cases

we must de¬ne the two classes as generalized degrees of freedom. To avoid

unnecessary accumulation of complexity, we shall in the following consider

cartesian coordinates ¬rst, and consider necessary modi¬cations resulting

from the use of generalized coordinates later (Section 8.6.1 on page 263).

In the case that the relevant coordinate is a distance between atoms or a

linear combination of atomic coordinates, the equations are the same as for

cartesian coordinates of selected particles, although with a di¬erent e¬ective

mass.

8.2 The generalized Langevin equation

Assume we have split our system into explicit “relevant particles” indicated

with a prime and with positions r i (t) and velocities v i (t), i = 1, . . . , N ,

and implicit double-primed “irrelevant” particles with positions r j (t) and

velocities v j (t), j = 1, . . . , N . The force F i acting on the i-th primed

particle comes partly from interactions with other primed particles, and

partly from interactions with double-primed particles. The latter are not

available in detail. The total force can be split up into:

• systematic forces F s (r ) which are a function of the primed coordinates;

i

these forces include the mutual interactions with primed particles and the

interactions with double-primed particles as far as these are related to the

primed positions;

• frictional forces F f (v) which are a function of the primed velocities (and

i

may parametrically depend on the primed coordinates as well). They

include the interactions with double-primed particles as far as these are

related to the primed velocities;

• random forces F r (t). These are a representation of the remainder of the

i

interactions with double-primed particles which are then neither related to

252 Stochastic dynamics: reducing degrees of freedom

the primed positions nor to the primed coordinates. Such forces are char-

acterized by their statistical distributions and by their time correlation

functions. They may parametrically depend on the primed coordinates.

This classi¬cation is more intuitive than exact, but su¬ces (with addi-

tional criteria) to derive these forces in practice. A systematic way to derive

the time evolution of a selected subsystem in phase space is given by the pro-

jection operator formalism of Zwanzig (1960, 1961, 1965) and Mori (1965a,

1965b). This formalism uses projection operators in phase space, acting

on the Liouville operator, and arrives at essentially the same subdivision of

forces as given above. In this chapter we shall not make speci¬c use of it.

The problem is that the formalism, although elegant and general, does not

make it any easier to solve practical problems.1

We make two further assumptions:

(i) the systematic force can be written as the gradient of a potential in

the primed coordinate space. This is equivalent to the assumption

that the systematic force has no curl. For reasons that will become

clear, this potential is called the potential of mean force, V mf (r );

(ii) the frictional forces depend linearly on velocities of the primed parti-

cles at earlier times. Linearity means that velocity-dependent forces

are truncated to ¬rst-order terms in the velocities, and dependence

on earlier (and not future) times is simply a result of causality.

Now we can write the equations of motion for the primed particles as

t

‚V mf

dv i

=’ ’ ζij („ )v j (t ’ „ ) d„ + · i (t),

mi (8.1)

dt ‚r i 0

j

where ζij („ ) (often written as mi γij („ )) is a friction kernel that is only

de¬ned for „ ≥ 0 and decays to zero within a ¬nite time. The integral

over past velocities extends to „ = t, as the available history extends back

to time 0; when t is much larger than the correlation time of ζij („ ), the

integral can be safely taken from 0 to ∞. This friction term can be viewed

as a linear prediction of the velocity derivative based on knowledge of the

past trajectory. The last term ·(t) is a random force with properties still

to be determined, but surely with

·(t) = 0, (8.2)

v i (t) · · j (t ) = 0 for any i, j and t ≥ t. (8.3)

1 van Kampen (1981, pg 398) about the resulting projection operator equation: “This equation

is exact but misses the point. [. . . ] The distribution cannot be determined without solving the

original equation . . . ”

8.2 The generalized Langevin equation 253

On the time scale of the system evolution, the random forces are stationary

stochastic processes, independent of the system history, i.e., their correlation

functions do not depend on the time origin, although they may have a

weak dependence on system parameters. In principle, the random forces are

correlated in time with each other; these correlations are characterized by

·

correlation functions Cij („ ) = · i (t)· j (t + „ ) , which appear (see below) to

be related to the friction kernels ζij („ ).

This is the generalized Langevin equation for cartesian coordinates. For

generalized coordinates {q, p} the mass becomes a tensor; see Section 8.6.1

on page 263.

Note At this point we should make two formal remarks on stochastic equations

(like (8.1)) of the form

dy

= f (y) + c·(t), (8.4)

dt

where ·(t) is a random function. The ¬rst remark concerns the lack of mathematical