2.2 Linear response
Thus far, our discussion has not distinguished between a single system
or an interacting system. A general framework for describing its
dynamics as well as the equilibrium properties of interacting systems
approaching equilibrium can be understood in light of the linear
response theory, which is the foundation of nonequilibrium
thermodynamics. A system at equilibrium evolving under a Hamiltonian H
experiences a perturbation ΔH=fA, where f is the field variable (such as
an external force), and A is the extensive variable (such as the
displacement) that is conjugate to the field. The perturbation throws
the system into a nonequilibrium state, and when the field is switched
off, the system relaxes back to equilibrium in accordance with the
regression process described by Onsager [13]:
ΔA(t) = (f/kBT) 〈ΔA(0) ΔA(t)〉 (Eq. 8)
where, ΔA(t) = A(t) - 〈A〉. The above identity holds under linear
response, when ΔH is small, or equivalently when ΔA(t, λf)= λ ΔA(t, f).
The most general form to relate the response A to the field f under the
linear response is given by: ΔA(t)=∫ς(t-t’)f(t’)dt’. Here, we have
further assumed that physical processes are stationary in the sense that
they do not depend on the absolute time, but only the time elapsed,
i.e., ς(t, t’)= ς(t-t’). One can use the linear-response relationship to
derive an equation for the dynamics of a system interacting with a
thermal reservoir of fluid (also called a thermal bath). For example,
the dynamics of the particle (in one-dimension along the x-coordinate
for simplicity of illustration is given by
md2U/dt2 = - dV(x)/ dx + f, U is the
particle velocity, where V(x) is the potential energy function, and f is
an external driving force including random Brownian forces from the
solvent degrees of freedom. The thermal bath will experience forces
fr in the absence of the particle, and when the particle
is introduced, the perturbation will change the bath forces to f. This
change f-fr can be described under linear response as:
Δf(t)=f-fr = ∫ςb(t-t’)x(t’)dt’. Using
this relationship, and by performing integration by parts, the particle
dynamics may be written as:
md2U/dt2 = - dV(x)/ dx +
fr - ∫ξb(t-t’)U(t’)dt’ (Eq. 9)
Here the subscript b stands for bath,
ςb(t)=-dξb/dt, and fr is
the random force from the bath that is memoryless. This form of the
equation for the dynamics of the interacting system is referred to as
the generalized Langevin equation, and it accounts for the
memory/history forces. We note that while the parent equation (i.e., the
master equation) is Markovian, the memory emerges as we coarse-grain the
timescales to represent the system-bath interactions and is a
consequence of the 2nd law of thermodynamics. One can
recover the Langevin equation from the GLE by assuming that the memory
function in the integral of (Eq. 9) is a Dirac delta function. The
strength of the random force that drives the fluctuations in the
velocity of a particle (as noted in the above example) is fundamentally
related to the coefficient representing the dissipation or friction
present in the surrounding viscous fluid. This is the
fluctuation-dissipation theorem [15]. The friction coefficient,
ξb, associated is time-dependent and not given by the
constant value (given by the Stokes formula or a drag coefficient). In
any description of system dynamics, and therefore, the mean and the
variance of observables under the thermal fluctuations have to be chosen
to be consistent with the fluctuation-dissipation theorem. In order to
achieve thermal equilibrium, the correlations between the state
variables should be such that there is an energy balance between the
thermal forcing and the dissipation of the system as required by the
fluctuation-dissipation theorem [15]. Finally, we note that the
fluctuation theorems of Crooks and the Jarzynski relationships for
relating equilibrium free energies to nonequilibrium work can be derived
from ratios of the probabilities of the forward and backward paths of a
Markov process [16].