Transcript The Local Free Energy Landscape

Internationa Research Training Group
Diffusion in Porous Materials
Workshop Leipzig 9th-12th May 2006
The Local Free Energy Landscape - a Tool
to Understand Multiparticle Effects
Siegfried Fritzsche
Leipzig 10th May 2006
University of Leipzig,
Institut of Theoretical Physics
Group Molecular Dynamics / Computer
Simulations
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
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.
Phase space formulation of Statistical Physics
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 of
N spherical particles with 3N degrees of freedom is given by
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
That means
The local configurational integral is
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 local free energy
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 valid
at arbitrary density.
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 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 Mechanics
that 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.
The potential of mean force
Consider the following quantity
That means
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.
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.
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
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.
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 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.
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 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 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 value
of 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 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!
A simple example
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 A1 = a and A2=2a and the potential energies U1=E and U2 = 2E .
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 is
in this case simply the constant density multiplied with the
cross 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 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.
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.
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.
Metropolis MC
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}.
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.
Umbrella sampling
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!
We ask now, what will happen, if we add another potential Ub to the
existing one. We can write without change in the result
or even
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 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.
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 r0
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+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 a
normalization.
How to normalize in the case of more than one
boost potential?
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.
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.
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
crosses: biased Ab=0.5
full circles: biased Ab = 0.05
stars: biased Ab=0.5, multiplied by 2.6
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.