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].