The Local Free Energy Landscape - a Tool to Understand Multiparticle Effects

University of Leipzig, Institut of Theoretical Physics

Group Molecular Dynamics / Computer Simulations

Siegfried FritzscheLeipzig 10th May 2006

Internationa Research Training Group Diffusion in Porous Materials

Workshop Leipzig 9th-12th May 2006

Contents

Aim of this talkPhase space formulation of statistical physicsThe local free energyThe potential of mean forceChandlers reversible work theoremThe local entropyThe free energy in the one dimensional projectionA simple exampleHow to obtain the local free energy from simulationsMetropolis Monte Carlo simulationsUmbrella samplingAn example: A spherical particle in a model potential

Aim of this talkIn 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 MechanicsOxford 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 makeyou more familiar with this stuff and its applications.

In statistical physics systems are described by a probability distribution ρN

in the phase space. The expectation value for a quantity A in a system ofN spherical particles with 3N degrees of freedom is given by

Phase space formulation of Statistical Physics

In the canonical ensemble ρN is given by

In the normalization factor the Q is called the partition function

The classical canonical partition function is

with the de Broglie wave length

The hamiltonian H is equal to the total energy

and ZN is the configurational integral

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

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 n0 is a constant normalization factor. This finding is validat arbitrary density.

The local free energy

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.) thatacts on each adsorbed particle separately:

This is the barometric law, where n0 is a normalization factor that depends only upon T. Hence,

Just to summarize:For the above defined local free energy it follows from Statistical Mechanicsthat it can be expressed by the local one particle density as:

This is valid for any density.

For low density we have

where is 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.

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 averageforce on a particle at arbitrary density that becomes the potential of theexternal forces in the limit of low densities.

What is the mean force?

Interpretation: Put particle 1 on site while the other particles are distributed randomly. Messure the force on particle 1 whichcomes from other particles, walls, external fields etc. Look at all possible such situations. Let each one of them appearwith the probability that they have in the canonical ensemble.

Average over all these situations to get the average force we speak about.

Example: S. Shinomoto, Phys. Lett. 89A (1982) 19

Equation of state derived from the „pressure“ on a hard sphere particlenear 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

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 distributionof the other particles - as it has to be.Therefore, the average force is just the force in the usual sense.

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.

Conclusions:

We have defined a local free energy and we have shown

1) how it is related with the single particle density2) that this local free energy is the potential of the mean force

In order to understand the behavior of a single particle in an ensembleof many particles one should consider the local free energy landscaperather than the potential landscape.

At low density the local free energy (in this full description, that includesall degrees of freedom) is the one particle potential energy.

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

If we express the one particle density by the local free energyit can now be factorized as

in an energetic factor and an entropic factor.

Note that in this description in the space of spherical particlesthe local entropy for a particle is only related to the influenceof the other particles and becomes a constant part of n0

at high dilution.

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 valueof the x – coordinate is

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 givesThe 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!

Consider the following system: A dilute gas is found in two volumes thatare connected. Let the potential energy and the cross section in yz – directions be constant in each subvolume. Let the cross sections e. g. be A1 = a and A2=2a and the potential energies U1=E and U2 = 2E .

A simple example

The particle density n(x) follows from the barometric law. n1(x,y,z) = n0 exp(-ßE), n2(x,y,z) = n0 exp(-2ßE),where n0 is a common normalization factor.

The one dimensional probability density p(x) as defined above isin this case simply the constant density multiplied with thecross section area.

p1(x) = a n0 exp(-ßE) p2(x) = 2a n0 exp (-2ßE)

The local free energy is Fi(x) = - kT ln pi(x) respectively.Let Fi be the constant value of F in region i. Then we have

F2 – F1 = E – kT ln 2

For low temperatures it is clearly F2 > F1 as to be expected from E2 > E1 .

But, with increasing temperature F2 – F1 becomes more and more negativeWhile the difference in the potential energy remains the same.

The reason for this effect is that the larger volume of region Bis now hidden in the definition of F(x) in the reduced one dimensional description.

Note, that in the 3d description the differences between local freeenergy and potential energy came from the contributions of otherparticles to the mean force. Now, in the reduced 1d description, theprojection creates additional contributions - even at high dilution.

Such an effect can also appear if, instead of, or additional to y, z angular degrees of freedom are also projected in the one remainingdimension. An example will be given below.

Conclusion: The local free energy landscape and the potential landscapecan look completely different. Physically more meaningful is thelocal free energy landscape as it is the potential of the mean force.

The one dimensional description makes it possible to examine complexphenomena of multidimensional systems along one importantcoordinate 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.

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 ourselvesto high dilution and three dimensions. The method is the same formany 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 than3 dimensions.

For d=5 and higher e.g. Monte Carlo (MC) simulation is muchmore effective.

Basic idea of Metropolis MC: do random shifts of your particle. LetΔU 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 withthe probability exp{-βΔU}.

Metropolis MC

Provided the walk is long enough and the systemis ergodic then the density of visited sites will converge against thedensity distribution n, but unnormalized of course. Normalization is theneasy by dividing by the number of all shifting events.

We ask now, what will happen, if we add another potential Ub to theexisting 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 theseregions are the mostly interesting ones.

Therefore, it would be desireable to „boost“ these regions, thatmeans 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

Let us call the original distribution the unbiased one

and the average with this distribution as

The new distribution is 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 ofmain interest i.e. where U has high values.

Then the first factor can easily evaluated with good statistics as the regionof interest is much more frequently visited.

The second factor however gives poor results because the boostpotential is small where the Boltzmann factor his high.

Fortunately, this factor is only a number that is common to all r0

Therefore, it drops out during the normalization.

We can resume: Instead of the original random walk we do anotherone in a potential landscape U+Ub 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 anormalization.

This is the most important problem in most of the applicationsof 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 theseregions overlap. Then they can be adjusted by multiplicationwith a constant for each of them to create one continousunnormalized 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 densitythen the local free energies for the different regions are shiftedby additive constants.

Adjusting the free energy from different subregions(arbitrary example curve)

The solid curve is created by shifting the dashed ones

A simple example: A spherical molecule in a model potential

U( x,y,z) = A x4 - Bx2 + C( y2+z2)

A = 5 10-3 kJ/mol B = 0.8 kJ/mol C = 20.0 kJ/mol

Potential energy along the x - axis

minimum at -32.0 kJ/mol, saddle point at 0.0 kJ/mol

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

The density distribution for higher loading

Solid line: density distribution at high dilution

The potential of mean force (one dimensional version)

Boost potentials Ub = Ab x2 are added, Ab = 0.25 and Ab = 1.0

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 (Ab=0.05)

The normalized density near the saddle point

triangles: unbiased full circles: biased Ab = 0.05crosses: biased Ab=0.5 stars: biased Ab=0.5, multiplied by 2.6

Conclusions:

Umbrella sampling improves the accuracy in the biased regionsconsiderably.

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 toa common factor that must be found somehow.