Failure of the work-Hamiltonian connection for free energy calculations
Failure of the work-Hamiltonian connection for free energy calculations 
Failure of the work-Hamiltonian connection for free energy calculations 
Jose M. G. Vilar1 and J. Miguel Rubi2 
1Computational Biology Program, Memorial Sloan-Kettering Cancer Center, 1275 York 
Avenue, New York, NY 10021 
2Departament de Fisica Fonamental, Universitat de Barcelona, Diagonal 647, 08028 
Barcelona, Spain 
Abstract 
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 H x t 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. 
PACS numbers: 05.40.-a, 05.20.-y, 05.70.Ln 
Hamiltonians provide two key ingredients to bridge the microscopic structure of nature 
with macroscopic thermodynamic properties: they completely specify the underlying 
dynamics and they can be identified with the energy of the system [1].  At equilibrium, 
the link with the thermodynamic properties is established through the partition function 
( )H xZ e dβ−= ∫ x , which here uses the Hamiltonian  in the coordinate space ( )H x x  as the 
energy of the system [2]. In particular, the free energy is given by 
= − Z , where 
1 Bk Tβ ≡ /  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. 
The connection between work and Hamiltonian expressed through the relation 
W H x
)t, , or equivalently  through its integral representation 
( ( ') ') '
W H x t t
∂∫ dt , is typically used to extend statistical mechanics to far-from-
equilibrium situations [3-5]. 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  
 0( ) ( ) ( ) (H x t H x f t x g t), = − + , 
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.  
( )g t
0 /F H x f= −∂ ∂ + ( )t ( )g t
( )g t
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 
( )g t
0x , is subjected to a sudden perturbation 0( ) ( )f t f t≡ Θ , where 0f  is a 
constant and  is the Heaviside step function. The work performed on the system, 
, where 
( )tΘ
0( tW f x x= − 0 ) ( )tx x t≡  represents the value of the coordinate x  at time , is in 
general different from 
' 0 0
( ') ' ( ) (0
tH x t dt f x g t gt
, = − + −
∂∫ ) , irrespective of the explicit 
form of the function .  ( )g t
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 
 ,EG We eβ β− Δ −=  
which have been widely used recently to obtain estimates EGΔ  of free energy changes 
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 We 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 revG WΔ = , where  
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 
( )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  
( )H x x−∂ /∂
0 0( ) ( )rev tW H x H x= − 0
  0 0( , ) ( ,0)rev eq t eq tG W P x t P x dx dxΔ = ,∫ ∫
where the equilibrium probabilities  are obtained, in the usual way, from the 
Boltzmann distribution 
( )( , )
H x t
eq Z tP x t e
β− ,= . To be explicit, let us consider a harmonic 
system described by 210 2( )H x kx=  and ( ) 0g t = , with  a constant. In this case, we can 
compute exactly the free energy change: 
G kxΔ = ,  
where ( )eqx f t k≡ / , 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))ZG Z t ZβΔ = − / , where 
( )( ) H x tZ t e dβ− ,= ∫ x  is 
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 212Z eqG kΔ = − x , a negative value inconsistent with a process that is not 
spontaneous. More generally, the Hamiltonian 212( ) ( )(tH x t kx f t x )γ, = − − , where γ  is a 
constant parameter that does not affect the dynamics of the system, leads to 
2( )Z eq eqG kx xγΔ = − , which can be positive or negative depending on the value of γ . 
Therefore, the estimates ZGΔ  are not suitable to predict typical thermodynamic 
properties, such as whether or not a process happens spontaneously. 
To what extent does the failure of the work-Hamiltonian connection impact 
nonequilibrium work equalities? In the case of a sudden perturbation and a harmonic 
potential discussed previously, the following result follows straightforwardly:  
 0 0( ) 0 0( , ) ( ,0) 1t
f x xW
eq t eq te e P x t P x dx dx
ββ − −− = =∫ ∫ ,  
which is different from .  Ge β− Δ
An intriguing question then arises: why do experiments and computer simulations 
sometimes lead to results that agree with nonequilibrium work equalities? Let us consider 
a situation closer to the experimental and computational setups, with a harmonic time-
dependent force that constrains the motion on the coordinate x :  
 210 2( ) ( ) ( )tH x t H x K x X, = + − .  
Here  is a constant and K tX  is the time-dependent equilibrium position for the 
constraining force. In this case, with 210 2( )H x kx=  and 0 0X = , we also have 
2rev eq
G W kxΔ = = ,  
where now Keq tk Kx X+≡ . 
For quasi-equilibrium displacements of tX , so that the work performed is equal to 
the reversible work,  , we have  0 0( ) ( )rev tW W H x H x= = − 0
 0 0 0( ( ) ( )) 0 0( , ) ( ,0)rev t
W H x H x
eq t eq te e P x t P x dx
β β− − −= ,∫ ∫ dx  
which leads to 
( ) 2
2( 2 ) ( )
k k K
eqk K
W e ke
K k K
This result indicates that quasi-equilibrium does not guarantee the accuracy of the 
exponential estimate of the free energy from nonequilibrium work relations. The free 
energy change  and its exponential estimate GΔ EGΔ  agree with each other only for large 
values of . The reason is that, in this case, work and Hamiltonian are connected to each K
other when both quasi-equilibrium and large-  conditions are fulfilled simultaneously. 
Under such conditions, the work-Hamiltonian connection is valid because 
eq tx x X≈ ≈  
implies that the rate of change of the Hamiltonian, ( ) / ( ) /t tH x t t K x X dX dt∂ , ∂ = − − , 
equals the power associated with the external force, / ( ) /tdW dt K x X dx dt= − − . 
Interestingly, large values of  suppress fluctuations and lead to quasi-deterministic 
dynamics. Indeed, the experimental data [6] and computer simulations [7] indicate that 
the agreement between the free energy change 
GΔ  and its exponential estimate EGΔ  
occurs mainly for relatively slow perturbations that lead to quasi-deterministic 
trajectories.  
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.  
References 
[1] H. Goldstein, Classical mechanics (Addison-Wesley Pub. Co., Reading, Mass., 
1980). 
[2] R. C. Tolman, The principles of statistical mechanics (Oxford University Press, 
London, 1955). 
[3] C. Jarzynski, Physical Review Letters 78, 2690 (1997). 
[4] G. Hummer, and A. Szabo, Proc Natl Acad Sci USA 98, 3658 (2001). 
[5] A. Imparato, and L. Peliti, Physical Review E 72, 046114 (2005). 
[6] J. Liphardt et al., Science 296, 1832 (2002). 
[7] S. Park et al., Journal of Chemical Physics 119, 3559 (2003). 
[8] J. Gore, F. Ritort, and C. Bustamante, Proc Natl Acad Sci USA 100, 12564 (2003). 
[9] D. Reguera, J. M. Rubi, and J. M. G. Vilar, Journal of Physical Chemistry B 109, 
21502 (2005). 
