FOR INTERNAL USE ONLY (CONFIDENTIAL)
AUTHOR:
Dr. Rolf Wüthrich, MIE
DATE:
30.04.2007
PROJECT:
Dynamical Ising model
Field of Applications




Thermodynamics out of equilibrium
Understanding of entropy production in non equilibrated systems
Magnetic nano-particles
Surface science (in particular to electrochemical processes)
Model description:
We consider a lattice of size L. On each lattice node 1  i  L we associate a generalized spin
 i which may have either that value  i  1 or  i  1 (a two-level system). The probability
that the spin at the node i is in the state  i  1 is Pi  i  1 .
The dynamics of the system is governed by following 2L master equations:

Pi  i t   1    Pi  i t   1   Pi  i t   1
t
(1)

Pi  i t   1   Pi  i t   1   Pi  i t   1
t
where  and  are the transition probabilities between the two states of the generalized spins.
We propose to write these transition probabilities as follows:
 E
 k BT



 E
 k BT



  vo1 exp 
(2)
  vo1 exp 
Department of Mechanical & Industrial Engineering
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 E
E 

0

Figure 1: Energy profile of the two-level system.
with Ei the difference of energy between the energy maxima (due to the energy barrier the
system has to pass in order to flip the spin) and the energy minima corresponding to the
equilibrium position (see figure 1). For a symmetric system E  E and therefore    .
Order parameter
We introduce following order parameter (generalized magnetisation):
(3) M 
1
 i   i
L i
If p is the fraction of spins in the state  i  1 , then the order parameter will be equal to:
(4) M  p  1  p  2 p 1
Entropy
The Gibbs entropy is defined as:
S  k B  P ln P 

with P  the probability to find the system in the state   1, 2 ,, L  .
Department of Mechanical & Industrial Engineering
1455, de Maisonneuve Blvd. West, Montreal, Quebec H3G 1M8
Tel: (514) 848-2424 ext. 3150 Fax: (514)-848-3175 Email: [email protected]
2
For the particular case where all probabilities are equals (this is the case in the thermal
equilibrium) we get directly the Boltzmann formula for the entropry:
S  kB ln 
where  is the number of states accessible to the system (the inverse of the constant
probability P  ).
Solutions of the model
No interaction between spins
In case of no interaction between the spins, the solution of (1) is straight forward (telegraph
noise equation). If p represents the fraction of spins in the state  i  1 , the solution of (1)
writes [1]:
(5) pt  


1  exp     t 
The system will tend to following equilibrium state:
(6) p S 


Therefore the order parameter will be (at equilibrium):
(7) M 
2
1

For a symmetric system we get:
(8) p S  0.5
and
(9) M  0
In both situations (not symmetric and symmetric) no phase transition takes place at a finite
value of the temperature T.
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Note that the non-symmetric case is mathematically equivalent to the mean-field theory of
absorption processes on metals (applications in heterogeneous catalyses and electrochemistry).
If  is the surface coverage of the adsorbed molecule and c bulk its concentration in the
homogenous phase (in the vicinity of the interface) then one can write:
d
 k f 1   c bulk  kb
dt
which solution at equilibrium is the so-called Langmuir isotherm ( T  const ):

 G 0 


 aexp  
1   kb
RT


kf
with a the chemical activity and G 0 the standard adsorption enthalpy (which is function of
the electrode potential in the case of an electrochemical process).
Interaction between spins
To take into account the interaction between the spins, we suppose that the energy is modified
according table 1 (J is the exchange integral).
Spin configurations
Energy
-J
+J
Table 1: Interaction energy between two spins
As an example let us consider a system of two spins only. Let us write down the transition
probability for the situation shown on figure 2:
 E  2 J 

(10)   vo1 exp   o
k BT 

In analogue way one can write down the other transition probabilities (table 2).
Department of Mechanical & Industrial Engineering
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Tel: (514) 848-2424 ext. 3150 Fax: (514)-848-3175 Email: [email protected]
4
Eo
E  2 J

0

Figure 2: Flipping a spin for a system of two spins with interaction
Spin flipping mechanism
Transition probability
 E  2 J 

vo1 exp   o
k BT 

 E  2 J 

vo1 exp   o
k
T
B


 E  2 J 

vo1 exp   o
k BT 

 E  2 J 

vo1 exp   o
k
T
B


Table 2: Transition probabilities
The Metropolis Algorithm
We know that the expectation value of an observable A can be written as:
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Ae 

e 
 Er
r
A
r
 Er
r
where Ar is the value of A for the state r. So given a system that has a discrete number of states,
we could, using a computer, calculate A for each state and weight these values by their
Boltzman factors to find the average A . This might be feasible for a system with a small
number of states, but of course not for a large system.
What if we decide to just sample some of the states? How would we decide which ones? This is
where the “Monte Carlo" part comes in. Named for the Mediterranean casino town, a Monte
Carlo method is any algorithm that involves a pseudorandom number generator.
One (bad) way of using random numbers would be to randomly pick a lot of states, measure A
for each of them, and weight these values of A by their Boltzman factors. We might get close to
the right answer if we sampled a lot of states, but we would spend a lot of time calculating A
for states that contribute very little to the real result (an Ising lattice at very high temperature is
unlike to be in the state with all spins pointing in one direction).
Instead of sampling a lot of states and then weighting them by their Boltzman factors, it makes
more sense to choose states based on their Boltzman factors and to then weight them equally.
This is known as the Metropolis algorithm, which is an importance sampling technique. One
pass through the algorithm is described here:
1. A trial configuration is made by randomly choosing one spin.
2. The energy difference of the trial state relative to the present state, E , is calculated.
3. If E  0 , the trial state is energetically favorable and thus accepted. Otherwise, a
random number 0    1 is generated, and the new state is only accepted if
  exp  E  .
References:
1. R. Wuthrich, “Spark Assisted Chemical Engraving – A stochastic modelling approach”,
PhD Thesis, Swiss Federal Institute of Technology Lausanne, thesis number 2776
(2003).
Department of Mechanical & Industrial Engineering
1455, de Maisonneuve Blvd. West, Montreal, Quebec H3G 1M8
Tel: (514) 848-2424 ext. 3150 Fax: (514)-848-3175 Email: [email protected]
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