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In the introductory Chapter 1, we mentioned a few examples of phase transitions
of physical systems, for example that of the ferromagnet...
However (con-
(consider the last row of Table 6.1), when we calculate the specific heat we obtain two
different expressions above and below the critical temperature and thus a dis-
discontinuity at T = Te...
Here specific singularities of the
specific heat, etc., occur at the phase transition point which are described by so-
called critical exponents...
If there are several peaks (corresponding to different minima
of 9(q)) we assume that only one state q = q0 is occupied...
) It is required that V{q) is invariant under all transformations of q which leave
the physical problem invariant...
Equation for the relaxation ofq(x)
«*-~ш FЛ92)
6.8 Phase Transition Analogy in Continuous Media: Space-Dependent Order Parameter 187
where q is now treated as function of space, x, and time, t...
G.2)
In a chemical reaction, F will be a function of the concentration of chemical
reactants...
G.9)
Incidentally this guarantees the existence of the inverse of A, i.e., the determinant
is unequal zero
det^+O...
To establish the connection between the present case
and the former one we want to secure the validity of the adiabatic technique...
Let us first assume that these parameters are chosen
in such a way that the q°'s represent stable values...
The points 1-3 will be fully taken into account by the method described in Sec-
Sections 7.7 to 7.11...
It may be that nature supports evolution by changing external parameters so that
the just-described switching process becomes effective in developing new species...
Therefore it is sometimes desirable to perform the
adiabatic elimination technique with the Fokker-Planck equation...
We then decom-
decompose U
U = U0 + q G.60)
7.7 Generalized Ginzburg-Landau Equations for Nonequilibrium Phase Transitions 207
with
0\
4 = [': • G-61)
\g.(*,t)l
Splitting the rhs of G.59) into a linear part, Kq, and a nonlinear part q we obtain
G.59) in the form
= g(q) + F(t)...
Since all stable modes are damped while the unstable modes are undamped,
we are safe that the operator djdt — As possesses an inverse...
To write the perturbation theory in the same form for both cases, we introduce
the concept of modified Green's functions...
Combining the expressions G.147),
G.150), G.151) we can write the final equation for the slowly varying amplitude
£(R, T) which reads
Щр - B-V2RZ(R, T) = (-C + M\ Z(R, T)\2)S,{R, T) + F(R, T)
G.155)
with А, В, С defined in G.148)...
Fx is a stochastic
force which occurs necessarily due to the unavoidable fluctuations when dissipation
is present...
We may safely
neglect the nonlinearity, and the field is merely supported by stochastic processes
(spontaneous emission noise)...
(8.33)
In the same approximation we find
*„ = Уц№ - °д) - 2 Ея »А», (8-34)
or, after solving (8.34) again adiabatically
D = £„ <т„ « Do - ^ Хя н-iiii, (8.35)
where Dc is the critical inversion of all atoms at threshold...
(8.90)
A closer investigation of (8.88), (8.89) reveals that Xm > 0 for Do > D^ and an
instability occurs, at first for k = 0...
If
finally d > 0, R = 0 becomes unstable (or, more precisely, we have a point of
marginal stability) and the new stable point lies at R — R2...
As long as the Rayleigh number is not too large, the fluid remains quiescent and
heat is transported by conduction...
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