Figure 5.13:

The equilibrium reaction extent's probability density for Reactions 5.52 at system volume \Omega =20 (top) and \Omega = 200 (bottom). Notice the decrease in variance in the reaction extent as system volume increases.

Code for Figure 5.13

Text of the GNU GPL.

main.m


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% Copyright (C) 2003, James B. Rawlings and David Q. Mayne
%
% This program is free software; you can redistribute it and/or
% modify it under the terms of the GNU General Public License as
% published by the Free Software Foundation; either version 2, or (at
% your option) any later version.
%
% This program is distributed in the hope that it will be useful, but
% WITHOUT ANY WARRANTY; without even the implied warranty of
% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
% General Public License for more details.
%
% You should have received a copy of the GNU General Public License
% along with this program; see the file COPYING.  If not, write to
% the Free Software Foundation, 59 Temple Place - Suite 330, Boston,
% MA 02111-1307, USA.

% Stochastic reaction simulation
% A +B <-> C
% Determine equilibrium via chemical master equation

% Set state vector x: [A;B;C]
Vol = 20;
x = [1;5;0]*Vol;
nmol = x(1);
% Set the reaction rate vector k
k(1) = 1/Vol;
k(2) = 3;
extent = [1:-1/nmol:0]';

% Construct A of the chemical master equation
% dP/dt = A*P
A = zeros(nmol+1,nmol+1);
for i = 1:nmol+1
  n = i-1;
  nu = nmol-n;
  a = n;
  b = x(2)-nu;
  c = nu;
  A(i,i) = -(k(1)*a*b + k(2)*c);
  if (i>1)
    A(i,i-1) = k(2)*(c+1);
  end
  if (i<nmol+1)
    A(i,i+1) = k(1)*(a+1)*(b+1);
  end
end

v = null(A);
v = v/sum(v);
figure()
plot(v)
%
% set initial condition for p
%
p = zeros(nmol+1,1);
p(nmol+1) = 1*nmol;

deltat = 0.01;
tfinal = 1;
time = [0:deltat:tfinal]';
Ad = expm(A*deltat);
for i = 1:rows(time)
  p(:,i+1) = Ad*p(:,i);
  p(:,i+1) = p(:,i+1)/sum(p(:,i+1))*nmol;
end
peq = p(:,end)/nmol;
var_small = peq'*(extent-(peq'*extent)).^2
table_small = [extent, peq];

% Set state vector x: [A;B;C]
Vol = 200;
x = [1;5;0]*Vol;
nmol = x(1);
% Set the reaction rate vector k
k(1) = 1/Vol;
k(2) = 3;
extent = [1:-1/nmol:0]';

% Construct A of the chemical master equation
% dP/dt = A*P
A = zeros(nmol+1,nmol+1);
for i = 1:nmol+1
  n = i-1;
  nu = nmol-n;
  a = n;
  b = x(2)-nu;
  c = nu;
  A(i,i) = -(k(1)*a*b + k(2)*c);
  if (i>1)
    A(i,i-1) = k(2)*(c+1);
  end
  if (i<nmol+1)
    A(i,i+1) = k(1)*(a+1)*(b+1);
  end
end

%
% set initial condition for p
%
p = zeros(nmol+1,1);
p(nmol+1) = 1*nmol;

deltat = 0.01;
tfinal = 1;
time = [0:deltat:tfinal]';
Ad = expm(A*deltat);
for i = 1:rows(time)
  p(:,i+1) = Ad*p(:,i);
  p(:,i+1) = p(:,i+1)/sum(p(:,i+1))*nmol;
end
peq = p(:,end)/nmol;
var_large = peq'*(extent-(peq'*extent)).^2
table_large = [extent, peq];

figure()
plot(table_large(:,1),table_large(:,2),'^')

figure()
plot(table_small(:,1),table_small(:,2),'^')

save "prob_dis_scale.dat" table_small table_large