Failure of the work-Hamiltonian connection for free energy calculations

Extensions of statistical mechanics are routinely being used to infer free energies from the work performed over single-molecule nonequilibrium trajectories. A key element of this approach is the ubiquitous expression dW/dt=\partial H(x,t)/ \partial t which connects the microscopic work W performed by a time-dependent force on the coordinate x with the corresponding Hamiltonian H(x,t) at time t. Here we show that this connection, as pivotal as it is, cannot be used to estimate free energy changes. We discuss the implications of this result for single-molecule experiments and atomistic molecular simulations and point out possible avenues to overcome these limitations.

is the inverse of the temperature T times the Boltzmann's constant .
Thermodynamic properties play an important role because they provide information that is not readily available from the microscopic properties, such as whether or not a given process happens spontaneously. , ∂ ∫ dt , is typically used to extend statistical mechanics to far-fromequilibrium situations [3][4][5]. These relations are meant to imply that the work W performed on a system is used to change its energy. The potential advantage of this type of approach is that it would allow one to infer thermodynamic properties even when the relevant details of the Hamiltonian are not known or when they are too complex for a direct analysis. Experiments and computer simulations can thus be performed to probe the microscopic mechanical properties from which to obtain thermodynamic properties.
Time-dependent Hamiltonians, however, provide the energy up to an arbitrary factor that typically depends on time and on the microscopic history of the system. Such dependence, as we show below, prevents this approach from being generally applicable to compute thermodynamic properties.
To illustrate how work and Hamiltonian fail to be generally connected, we consider a system described by the Hamiltonian under the effects of a time-dependent force 0 ( ) H x ( ) f t . The total Hamiltonian is given by where is an arbitrary function of time, which leads to a total force . The function does not affect the total force but it changes the Hamiltonian. Therefore, has to be chosen so that the Hamiltonian can be identified with the energy of the system.
In general, the arbitrary time dependence of the Hamiltonian, , cannot be chosen so that the Hamiltonian gives a consistent energy. Consider, for instance, that the system, being initially at f is a constant and is the Heaviside step function. The work performed on the system, , where To illustrate the consequences of the lack of connection between work and changes in the Hamiltonian, we focus on the domain of validity of nonequilibrium work relations [3] of the type from single-molecule pulling experiments [6] and atomistic computer simulations [7].
The promise of this type of relations is that they provide the values of the free energy from irreversible trajectories and therefore do not require equilibration of the system. Yet, in almost all instances in which this approach has been applied, the agreement with the canonical thermodynamic results has not been complete and in some cases the discrepancies have been large. These discrepancies have been attributed to the presence of statistical errors in the estimation of the exponential average W e β − [8].
Currently, the mathematical validity of these type of nonequilibrium work relations appears to be well established: they have been derived using approximations [3] and rigorously for systems described by Langevin equations [4,5]. However, all these derivations rely in different ways on the work-Hamiltonian connection, which as we show below prevents them from giving general estimates of thermodynamic free energies.
The free energy difference between two states is defined as is the work required to bring the system from the initial to the final state in a reversible manner [2]. Note that, if the system is not macroscopic, is in general a fluctuating quantity. At quasi-equilibrium, the external force rev W rev W ( ) f t balances with the system force . After integration by the displacement, the reversible work done on the system is given by . Therefore, the free energy follows from . To be explicit, let us consider a harmonic system described by and ( ) 0 g t = , with a constant. In this case, we can compute exactly the free energy change: / , which leads to a positive value as required for non-spontaneous processes.
One might have been tempted to use the partition function to estimate changes in free energy according to the expression 1 ln( ( ) (0)) the time-dependent quasi-equilibrium partition function [3,4]. However, this relation is not valid when changes in the Hamiltonian cannot be associated with changes in energy.
In the case of the harmonic potential, the use of the time-dependent partition function leads to Here is a constant and K t X is the time-dependent equilibrium position for the constraining force. In this case, with 2  Bringing thermodynamics to nonequilibrium microscopic processes [9] is becoming increasingly important with the advent of new experimental and computational techniques able to probe the properties of single molecules [6,7]. Our results show that the classical connection between work and changes in the Hamiltonian cannot be applied straightforwardly to time-dependent systems. As a result, quantities that are based on the work-Hamiltonian connection, such as those obtained from nonequilibrium work relations and time-dependent partition functions, cannot generally be used to estimate thermodynamically consistent free energy changes. A possible avenue to overcome these limitations, as we have shown here, is to identify the particular conditions for which work and changes in the Hamiltonian are connected to each other.