1. The Local Free Energy Landscape - a Tool to Understand Multiparticle Effects University of Leipzig, Institut of Theoretical Physics Group Molecular Dynamics / Computer Simulations Siegfried Fritzsche Leipzig 10th May 2006 Internationa Research Training Group Diffusion in Porous Materials Workshop Leipzig 9th-12th May 2006

2. Contents Aim of this talk Phase space formulation of statistical physics The local free energy The potential of mean force Chandlers reversible work theorem The local entropy The free energy in the one dimensional projection A simple example How to obtain the local free energy from simulations Metropolis Monte Carlo simulations Umbrella sampling An example: A spherical particle in a model potential

3. Aim of this talk In nearly all papers about transition state theory the notion of the Local Free Energy is used but, rarely explained in detail what it really means. The notion of the Local Free Energy was introduced in the paper J. Chem Phys. 68 (1978) 2959 An short explanation is given in Chandlers book: David Chandler, Introduction to Modern Statistical Mechanics Oxford University Press, New York, 1987 But rigorous derivations are not given in this book. The present talk has the intention to fill this gap and to make you more familiar with this stuff and its applications.

4. In statistical physics systems are described by a probability distribution ρ N in the phase space. The expectation value for a quantity A in a system of N spherical particles with 3N degrees of freedom is given by Phase space formulation of Statistical Physics In the canonical ensemble ρ N is given by In the normalization factor the Q is called the partition function

5. The classical canonical partition function is with the de Broglie wave length The hamiltonian H is equal to the total energy and Z N is the configurational integral

6. The well known one particle probability density is obtained by integrating the N – particle density over all but one degrees of freedom. This can be written as Analogously a local canonical partition function can be defined The local configurational integral is That means The prefactor in Q (which is important only at very low temperatures), depends only upon the temperature not upon the site. With that Z the local density can be written as

7. The common definition of the Helmholtz free energy is Analogously a local free energy can be defined where the constant does not depend upon the site. Hence, the density can be expressed as where n 0 is a constant normalization factor. This finding is valid at arbitrary density. The local free energy

8. Comparison with the barometric law: At high dilution the interaction between the adsorbed particles can be neglected. W reduces to the external potential (walls, gravity, etc.) that acts on each adsorbed particle separately: This is the barometric law, where n 0 is a normalization factor that depends only upon T. Hence,

9. Just to summarize: For the above defined local free energy it follows from Statistical Mechanics that it can be expressed by the local one particle density as: This is valid for any density. For low density we have whereis the potential energy of a single particle. It seems that the local free energy defined above is a quantity that can be used for some purposes instead of the potential energy. The analogy between Ф and F is even more fundamental. Let us consider the mean force on a particle at higher densities.

10. Consider the following quantity That means The potential of mean force But, is the force on particle 1 if particles 2,…,N are at positions,therefore, Hence, is the potential of the average force on a particle (Kirkwood 1935). Hence, the local free energy is nothing than the potential of the average force on a particle at arbitrary density that becomes the potential of the external forces in the limit of low densities.

11. What is the mean force? Interpretation: Put particle 1 on site while the other particles are distributed randomly. Messure the force on particle 1 which comes from other particles, walls, external fields etc. Look at all possible such situations. Let each one of them appear with the probability that they have in the canonical ensemble. Average over all these situations to get the average force we speak about.

12. Example: S. Shinomoto, Phys. Lett. 89A (1982) 19 Equation of state derived from the „pressure“ on a hard sphere particle near the wall that produces an effective potential of a mean force toward the wall. S1: uniform density assumed S2: pair correlation function is taken into account

13. The mean force at low density If the interactions between the moving particles can be neglected then In this case the average force does not depend upon the distribution of the other particles - as it has to be. Therefore, the average force is just the force in the usual sense.

14. Chandlers reversible work theorem The reversible work to move a particle from a site 1 to a site 2 is just the difference of the local free energy at the two sites. This follows immediately from the derivations given above. It is That means This is Chandlers reversible work theorem.

15. Conclusions: We have defined a local free energy and we have shown 1) how it is related with the single particle density 2) that this local free energy is the potential of the mean force In order to understand the behavior of a single particle in an ensemble of many particles one should consider the local free energy landscape rather than the potential landscape. At low density the local free energy (in this full description, that includes all degrees of freedom) is the one particle potential energy.

16. The local entropy A local entropy can also be defined by With the definition of the local partition function we find is the average total energy of the system if one particle is fixed at Instead of the well known formula we have in the local description with

17. If we express the one particle density by the local free energy it can now be factorized as in an energetic factor and an entropic factor. Note that in this description in the space of spherical particles the local entropy for a particle is only related to the influence of the other particles and becomes a constant part of n 0 at high dilution.

18. The free energy in the one dimensional projection In many cases a one dimensional description is desirable e. g. along the transition path crossing a saddle point in the free energy landscape. Therefore, often a one dimensional description is introduced by The probability p(x) to find a given particle to have a given value of the x – coordinate is

19. The local free energy can now be defined as Note, that although (because of the logarithm)can not be obtained by integration from A derivation completely analogous to the three dimensional one gives The average x – component of the force on a particle at site x Hence,is the potential of this mean force along the x dirction. In the limit of low density we have The low density limit of F(x) is NOT the potential nor its projection!

20. Consider the following system: A dilute gas is found in two volumes that are connected. Let the potential energy and the cross section in yz – directions be constant in each subvolume. Let the cross sections e. g. be A 1 = a and A 2 =2a and the potential energies U 1 =E and U 2 = 2E. A simple example The particle density n(x) follows from the barometric law. n 1 (x,y,z) = n 0 exp(-ßE), n 2 (x,y,z) = n 0 exp(-2ßE), where n 0 is a common normalization factor. The one dimensional probability density p(x) as defined above is in this case simply the constant density multiplied with the cross section area.

21. p 1 (x) = a n 0 exp(-ßE) p 2 (x) = 2a n0 exp (-2ßE) The local free energy is F i (x) = - kT ln p i (x) respectively. Let F i be the constant value of F in region i. Then we have F 2 – F 1 = E – kT ln 2 For low temperatures it is clearly F 2 > F 1 as to be expected from E 2 > E 1. But, with increasing temperature F 2 – F 1 becomes more and more negative While the difference in the potential energy remains the same. The reason for this effect is that the larger volume of region B is now hidden in the definition of F(x) in the reduced one dimensional description. Note, that in the 3d description the differences between local free energy and potential energy came from the contributions of other particles to the mean force. Now, in the reduced 1d description, the projection creates additional contributions - even at high dilution.

22. Such an effect can also appear if, instead of, or additional to y, z angular degrees of freedom are also projected in the one remaining dimension. An example will be given below. Conclusion: The local free energy landscape and the potential landscape can look completely different. Physically more meaningful is the local free energy landscape as it is the potential of the mean force. The one dimensional description makes it possible to examine complex phenomena of multidimensional systems along one important coordinate in a simple way. This is very an important advantage e. g. in Transition State Theory. An example is given in the talk about Transition State Theory.

23. How to obtain the local free energy from simulations? We can derive it from the density. But, where to get this density? A well known method is the umbrella sampling. We restrict ourselves to high dilution and three dimensions. The method is the same for many dimensions. We start from In d=3 dimensions the first part of this equation is already all what we need. But, the integral in the denominator is expensive to evaluate in more than 3 dimensions. For d=5 and higher e.g. Monte Carlo (MC) simulation is much more effective.

24. Basic idea of Metropolis MC: do random shifts of your particle. Let ΔU be the difference between the U of the new site and that of the old site. When exp{-βΔU} is larger than 1, then the trial move is always accepted. If exp{-βΔU} is smaller than 1, then the move is accepted with the probability exp{-βΔU}. Metropolis MC Provided the walk is long enough and the system is ergodic then the density of visited sites will converge against the density distribution n, but unnormalized of course. Normalization is then easy by dividing by the number of all shifting events.

25. We ask now, what will happen, if we add another potential U b to the existing one. We can write without change in the result For Metropolis MC a problem appears if the potential landscape includes regions of high potential energy. Then these regions are rarely visited by the random walk and the statistics is poor there. For many applications like transition state theory (TST) just these regions are the mostly interesting ones. Therefore, it would be desireable to „boost“ these regions, that means to do something to find the system more often in these states. This is done by the use of a so called boost potential! or even Umbrella sampling

26. Let us call the original distribution the unbiased one and the average with this distribution as The new distributionis called the biased one and the average with this distribution is written as Then we have Now we chose the boost potential so that it has low values in the region of main interest i.e. where U has high values. Then the first factor can easily evaluated with good statistics as the region of interest is much more frequently visited.

27. The second factor however gives poor results because the boost potential is small where the Boltzmann factor his high. Fortunately, this factor is only a number that is common to all r 0 Therefore, it drops out during the normalization. We can resume: Instead of the original random walk we do another one in a potential landscape U+U b and we calculate the average of instead of the average of Thats all. In practice however, one uses in most cases different boost potentials for different regions of the system. Then, the factors have different values for the different boost potentials that do not cancel out during a normalization.

28. This is the most important problem in most of the applications of umbrella sampling Each simulation with one of the boost potentials gives an unnormalised density with good accuracy for one region. The boost potentials are chosen in such a way that these regions overlap. Then they can be adjusted by multiplication with a constant for each of them to create one continous unnormalized density that finally can be normalized. How to normalize in the case of more than one boost potential? If one considers the local free energy rather than the density then the local free energies for the different regions are shifted by additive constants.

29. Adjusting the free energy from different subregions (arbitrary example curve) The solid curve is created by shifting the dashed ones

30. A simple example: A spherical molecule in a model potential U( x,y,z) = A x 4 - Bx 2 + C( y 2 +z 2 ) A = 5 10-3 kJ/mol B = 0.8 kJ/mol C = 20.0 kJ/mol

31. Potential energy along the x - axis minimum at -32.0 kJ/mol, saddle point at 0.0 kJ/mol

32. Energy distribution during MC runs at T = 200, 300, 400, 500 K

33. The density distribution for higher loading Solid line: density distribution at high dilution

34. The potential of mean force (one dimensional version)

35. Boost potentials U b = A b x 2 are added, A b = 0.25 and A b = 1.0

36. The normalized density in the one dimensional description from unbiased and biased MC runs The dashed line is the analytical solution (barometric law) triangles: unbiased MC full circles: biased MC (A b =0.05)

37. The normalized density near the saddle point triangles: unbiased full circles: biased A b = 0.05 crosses: biased A b =0.5 stars: biased A b =0.5, multiplied by 2.6

38. Conclusions: Umbrella sampling improves the accuracy in the biased regions considerably. Strong biasing leads to bad values in other regions. This leads to difficulties in the normalisation. The values in the biased region are even then correct up to a common factor that must be found somehow.