Calculating The Free Energy By The Jarzynski Equality

As one of the most important thermodynamic characteristic functions,the free energy plays an important role in a broad spectrum of applications.Nevertheless,to efficiently measure and calculate the free energy of a complex multi-body interacting system is still challenging.In experiments,the maximum work done by the system,and thus the free energy difference,can only be measured in the reversible thermodynamic process.However,any process in reality cannot be kept quasistatic.Theoretically,the calculation of the free energy involves the partition function,which cannot be solved analytically in general,and difficult to be calculated directly by numerical simulations.A conventional method for computing the free energy difference is the thermodynamic integration method,which needs to compute some related state variables as a function of the medium equilibrium states of the quasistatic process and thus the cost could be demanding.The Jarzynski equality provides a new theoretical basis to measure and calculate the free energy difference.According to the Jarzynski equality,the free energy difference between two given equilibrium states can be expressed by the statistical average of the work done in the nonequilibrium process.However,small work with rare probability weights heavily for the average,a hefty sample could be needed to evaluate it accurately,and thus is computationally more expensive and inefficient.As a consequence,to develop more efficient algorithms for the free energy calculation is still an important task in thermodynamics statistical physics.The main motivation of this thesis is to further study the Jarzynski equality and find a more effective way for the free energy calculation.According to the definition of the free energy,if two systems share the same Hamiltonian at two equilibrium states,respectively,then they share the same free energy difference between these two equilibrium states as well.Taking advantage of this flexibility,the free energy difference of a given system between two equilibrium states may be facilitated by considering the Jarzynski equality with instead another virtual system.The main task is therefore to explore the virtual system designed to this end.We show that by introducing an integrable virtual system,the evolution problem involved in the Jarzynski equality can be solved and thus the computational cost can be significantly reduced.In particular,we focus on the free energy difference of a given system under two different volumes.To this end,an applicable option is to take the hard wall cell potential in the virtual system,so that the free energy difference can be expressed in the form of an equilibrium equality.As a consequence,the free energy difference can be calculated by sampling the canonical ensemble directly.The effectiveness and efficiency of this scheme are illustrated with two models that represent lattices and fluids,respectively.The computational strategy of applying the Jarzynski equality to the virtual system can also be applied to the free energy difference between two values of any other parameters.The results obtained in this thesis may deepen our understanding to the free energy and provide more optional tools for the calculation of the free energy.