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We already know the theory of fluctuations of one variable . Just a reminder...
Professor: Do you know this?Students: No.
Professor: Did you know this last semester?
Students: No.
Professor: Were you taught this last semester?
Students: Yes.
Illustration to the Method of Successive Iterations
Suppose we have one variable x. Near equilibrium we construct the
thermodynamic potential A(x) for a subsystem: the free energy
of the subsystem at given x. The total free energy of the subsystem
plus the environment is minimal:
. We use Lagrange multiplier and
obtain
Probability of a fluctuation is determined by the entropy of the whole system:
![]() |
(1) |
We obtained:
Averages are:
Free energy:
![]() |
(2) |
Suppose we have many variables xi, i=1, 2,..., n. Let the equilibrium value
![]() |
(3) |
Let
Let us make a linear transformation:
xi = tik yk
Then
Mathematicians prove, that symmetric matrix can be
diagonalized: there exists matrix
such as
The quadratic form (3) becomes
Let us calculate :
© 1997 Boris Veytsman and Michael Kotelyanskii