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To thermodynamics and kinetics of second order phase transitions
14 August 2000
Physics Letters A 273 Ž2000. 61–69 www.elsevier.nlrlocaterpla
To thermodynamics and kinetics of second order phase transitions Yu.L. Klimontovich Department of Physics, M.V. LomonosoÕ Moscow State UniÕersity, Vorob’eÕy gory, 119899 Moscow, Russia Received 21 July 1997; received in revised form 10 April 2000; accepted 11 July 2000 Communicated by V.M. Agranovich
Abstract The accounts of thermodynamic and kinetic characteristics at second order phase transitions in ferroelectrics and antiferroelectrics for all temperature in critical region are carried out. Basis for account serves the kinetic equation for a local distribution function of values of a polarization vector Žthe order parameter.. The comparison of theoretical results with experimental date is considered. q 2000 Elsevier Science B.V. All rights reserved. Keywords: Second order phase transitions; Thermodynamics and kinetics; Self-consistent approximation for the first and the second moments; Critical region
1. Introduction In the ‘fluctuation theory of the second order phase transitions’ the critical point in the thermodynamic limit in mathematical sense is particular point. In this point the generalized static susceptibility, correlation radius of the simultaneous correlator of the order parameter fluctuations and correlation time of fluctuations tend to infinity. Such behavior of thermodynamic functions is a consequence of the thermodynamic limit transition w1–8x. Such behavior contradicts, however, to spirit both of thermodynamics and kinetics, which are based on model of continuous medium. In w7,8x for different levels of description the concrete definition of physically infinitesimal scales, the size of a point of continuous medium are given. E-mail address: [email protected] ŽY.L. Klimontovich..
That allows to enter natural small parameter – a ratio of the physically infinitesimal lengths l ph , determining the size of ‘point’ of continuous medium, to the characteristic size of system L.. The parameter l ph rL for a continuous medium approximation is always small. The expansion on parameter l ph rL was used for the Boltzmann gas and the Debye plasma w8,9x. As a result the generalized kinetic equations were constructed. The additional dissipative terms by the spatial diffusion of the corresponding distribution functions are defined. The unified description of kinetic and hydrodynamic processes without of the perturbation theory on Knudsen number is described. The principal difficulties of the Knudsen perturbation theory are set aside. The efficiency of expansion upon an parameter l ph rL is shown not only for Boltzmann gas, Debye plasma and the nonlinear Brownian motion, but for the theory of the second order phase transitions. The
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Yu.L. KlimontoÕichr Physics Letters A 273 (2000) 61–69
possibility arises to avoid ‘the problems of infinity’ – the thermodynamic and fluctuation characteristics are finite at all temperatures in the critical region. Such results corresponds to mathematical definition of the role of ‘infinity’, which with limiting clearness by Hilbert was formulated in his paper AOn the infinityB w10x: G the mathematical literature, if one more attentiÕe by look at it, abounds with nonsenses and absurds, for which an infinity is guilty in the most cases.H G Our general conclusion is as follows: in the realized kind infinity does not meet anywhere.H For the greater clearness of description an example of the elementary model of ferroelectrics and antiferroelectrics w11–14x are using. The results compare with the Landau theory w1– 4,13,14x. It is justified by that already at this level the alternative approach allows to enter the certain improvements in the theory of phase transitions of the second order. Despite of development of the more general ‘the fluctuation theory of phase transitions’, the Landau theory remains by powerful tool for the description of phase transitions in ferroelectrics w3,11,15x.
2. Landau and Boltzmann distributions. Thermodynamic functions 2.1. The Landau distribution function The theory of phase transitions developed by Landau, is a phenomenological: the exact effective Hamilton function Heff Žh ,T ., connected with the Hamilton function of system w13x, is replaced by model expression w8x. The most probable value hm .p for the distribution function f Žh ,t . plays a role of the order parameter at phase transitions. The effective Hamilton function Heff Žh ,T . is the additive characteristic and consequently is proportional to number of particles of system N or, accordingly, of the volume of system V: Heff Ž h ,T . s Nh eff Ž h ,T . .
Ž 1.
Here is entered designations h eff Žh ,T . for the effective Hamilton function in the account on one particle.
Replacing the exact effective Hamilton function by model expression, let’s receive ‘the Landau distribution’ for values of the order parameter f L Ž h ,T . s exp
N Ž c Ž T . y h eff Ž h T . . k BT
,
Ž 2.
where c ŽT . is the free energy in account on one particle. In a case of one the order parameter Žsee Ž144.3. in w2x. we write the following expression: h eff Ž h ,T . s
a T y TC 2
TC
b h 2 q h 4 y h F0 , 4
Ž 3.
where F0 is an external force connected to the order parameter h. At a zero external field from a condition of a minimum of function Heff Žh ,T . we find the most probable values hm .p , values of the order parameter:
hm .p s 0 hm .p s
at
T G TC ;
a T? y T b
TC
at
T F TC .
Ž 4.
The appropriate values of a susceptibility Žlinear response. is defined by the well-known formulas. From these formulas follows, that the order parameter in the critical point is equal to zero. It is impossible, however, to confirm this, as a susceptibility in the critical point tends to infinity and, hence, the critical point is unattainable. In the Gaussian 2 approximation the dispersion ² Ž dh . : is proportional to the factor 1rN and in a thermodynamic limit, at T / TC is equal to zero. 2.2. The Boltzmann distribution function To give the description of fluctuations, we replace the Landau distribution by the Boltzmann one f B Ž h ,T . s exp
Ž c Ž T . y heff Ž h T . . k BT
.
Ž 5.
It differs from the Landau distribution by absence of the factor N in the exponent. The account of thermodynamic functions on a basis of the Boltzmann distribution allows to carry out the account of thermodynamic functions in the critical region at all
Yu.L. KlimontoÕichr Physics Letters A 273 (2000) 61–69
values of temperature. For a concrete accounts the elementary model of ferroelectrics uses below: Heff Ž X ,T . ' Nh eff Ž X ,T . , h eff s
m v 02 2
X2
T y TC TC
q 12 bX 2 .
Ž 6.
The variable X defines the vector of polarization: P s enX. The most probable value of displacement Xm .p will play a role of the order parameter. b is the dissipative coefficient of nonlinearity. 2.3. The heat capacity of equilibrium and nonequilibrium states At the account C a by the distribution function f Ž X,a,T . with h eff Ž X,T . we can submitted the heat capacity as the sum of two contributions w13x C a s yT
E 2F
sT
E Seff
E Fth
. Ž 7. ET ET ET Thus, the heat capacity it is represented as sum of entropy and thermal force. It is consequence that the effective Hamilton function depends on temperature. In the Landau theory the account of thermodynamic functions is based on use a effective Hamilton function at the most probable value of internal parameter. At temperatures below critical are two symmetric values of the order parameter. The minima are divided by a barrier, height of which depends on temperature and aspires to zero at approach to the critical point. At phase transitions in the Landau theory for an nonsymmetric state the heat capacity, strictly speaking, corresponds to nonequilibrium state. On to measure of approach to the critical point the condition of the quasistatic state is broken. In result, appears the necessity for the definition of the heat capacity for nonequilibrium states. Such definition is possible to enter as follows. The kinetic equation for a local distribution function f Ž X. R.t . contains temperature as parameter and enables to find nonequilibrium distribution of values of the polarization – the function f Ž X. R.t ., at various values of temperature. It allows to enter the concept of free energy at a nonequilibrium condition w7,8x: 2
q
F Ž t < T . s² Heff :t y NS Ž t . .
The average effective energy and the entropy for nonequilibrium conditions, corresponding to given value of the temperature, are defined through the solution of the kinetic equation. Using definitions of free energy of equilibrium and nonequilibrium states it is possible to enter appropriate the Lyapunov functional:
L F s F Ž t < T . y FŽ o. Ž T . f Ž X . R.t . dR s k B Tn ln f Ž X . R.t . dX G 0. fŽ0. Ž X . V
H
Ž 9. With the help of the kinetic equation is established the second inequality d LF d Ž F Ž t < T . y FŽ o . Ž T . . s F 0. Ž 10 . dt dt The existence of the Lyapunov functional testifies the stability of equilibrium state, which at all values of temperature is defined by the Boltzmann distribution. The quasistatical states are unstable. The time of relaxation essentially depends on affinity to the critical point. The definition of nonequilibrium free energy, including the dependence from temperature as the parameter, allows to enter and definition of the heat capacity of a nonequilibrium state E 2 F Ž t
Ž 8.
63
m v 02 Ž TC y T . 4
bTC2
2
at
T - TC .
Ž 12 .
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64
From here follows, that the height of a potential barrier, dividing of two the most probable states, is proportional N and, hence, is infinite in the thermodynamic limit. The corresponding Kramers time tends to infinity. For Boltzmann distribution the depth of a potential hole is finite and is defined by the internal characteristic parameter of system ´ T s x T2 b. In the affinity to the critical point it is remained finite. Finiteness of the height of a barrier at any values of temperature is one of advantages of the Boltzmann distribution. In the Landau theory the heat capacity is defined by: C L Ž x m .p .,T . s Tc
E SL ET
sN
m v 02 2 bTc
'
1 kN
´T 2
.
ple, in magnetic alloys w17x and in high-temperature superconductors w18x.
3. Kinetic theory of the second order phase transitions In w8,13x proposed the following kinetic equation for the local distribution function f Ž X, R,t .:
E f Ž X , R ,t . Et E s
EX
DŽ x .
Ef EX
q
1 E h eff Ž X ,a f .
EX
mg
f qD
E R2
.
Ž 14 .
Ž 13 .
This expression defines the jump of the heat capacity. This result shows, that the heat capacity approach to the critical point from different levels. The critical point in framework of the Landau theory is unattainable. The Landau definitions of the entropy and the heat capacity do not coincide with results under the Gibbs theory. It gives the basis to speak about ‘the Landau thermodynamics’. The Landau results give only the part of the Gibbs heat capacity, which is defined by thermal force in the point X s X m .p w13x. From experimental data are known that two levels of the heat capacity Žbelow and above of the critical point. are connected by smooth curve with a finite height of a peak. The analytical account can be carried out only for small deviations from the most probable values. In this case it is possible to replace the Boltzmann distribution by the Gaussian one. The temperature dependence with in taken into account as the entropy, so and the thermal force at all values of temperature it is possible to obtain only on the basis of the Boltzmann distribution w13x. Results of numerical account at different values of the parameter ´ T are given on a Fig. 1 in w16x and w8x, section V.II. The height of the peak of the heat capacity temperature dependence is finite in the critical point at all values of temperature. A maximum of curves are displaced from a critical point in the side of low temperatures. Such behavior is observed, for exam-
E 2f
The kinetic equation contains two diffusion coefficients: of the spatial diffusion D s k B Trmg characterizes the resolve of nonhomogeneity in the position the center of dipoles; in the space of relative displacement of ions X. The second of them is defined by the formula DŽ x . s DrN for the Landau theory. For the Boltzmann distribution DŽ x . s D: Ds
k BT mg
s G X T2 ;
Gs
v 02 g
,
X T2 s
k BT m v 02
.
Ž 15 .
3.1. The self-consistent equation for the first moment From the kinetic equation follows the hooked chain of the equations for moments ² X n :. In the self-consistent approximation for the first moment the distribution function f Ž X , R ,t . s d Ž X y X Ž R ,t . . , ² X :s X Ž R ,t . .
Ž 16 . The equation for the function X Ž R,t . has a form:
E X Ž R ,t . Et
sy
v 02 T y Tc g
qD
Tc
q bX 2 Ž R ,t . X Ž R ,t .
E 2 X Ž R ,t . E R2
.
Ž 17 .
This equation has the structure of a reaction-diffusion equation. Such equations serve by a base in the
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65
theory of autowaves and autosolitons, and synergetics. The similar equation is used and in the theory of phase transitions w1–8x. It is named ‘Relaxation equation Ginsburg–Landau’ ŽREGL..
The reaction-diffusion equation can be established independently from the definition of the Landau functional Heff . The right part of the reaction-diffusion equation can be presented in form of a functional gradient from
3.2. REGL in Landau theory of phase transitions
E X Ž R ,t .
Following the Landau theory, we shall generalize expression for effective Hamilton function with taken into account the spatial distribution of the order parameter. It is corresponds to introduction of the appropriate Landau functional w2–6x, see also w8x, section V.II. For considered model it can be presented as: Heff s nm v 02
H
½
1 2
g EX q 2
T y Tc
X2
Tc
ER
Et
sy
Ueff s
d Heff
nm v 02
d X Ž R ,t .
H
.
Ž 19 .
Here is entered a phenomenological dissipative coefficient g .
½
Et
qg g
E R2
.
Ž 20 .
l
This equation only by designations Žg v 02rg ; g G D . differs from the reaction-diffusion equation. Thus the functional Heff plays in the Landau theory the double role: the role of the nonequilibrium potential in the REGL; the role of the effective Hamilton functional in appropriate the Gibbs distribution for the equilibrium state. These two roles, in general, are inconsistent.
l
Tc
q 12 bX 2
2
ž /5 ER
s2 DyG
q bX 2 Ž R ,t . X Ž R ,t .
E 2 X Ž R ,t .
2
dR.
Ž 22 .
For a polidomain and an antiferroelectric structures average meaning of the polarization vector is equal zero. More effective in this case is the selfconsistent approximation on second moment. The equation for the function EŽ R,t . s ² X 2 : has the form:
Et Tc
g
3.3. The self-consistent approximation for the second moment
E E Ž R ,t .
T y Tc
Ž 21 .
Here D is the coefficient of spatial diffusion.
E X Ž R ,t .
s yg
v 02 X 2 T y Tc
2
Ž 18 .
.
The nonequilibrium functional is defined by expression
D EX
dR.
g
d X Ž R ,t .
q
Here n is average density of number of particles. The factor g characterizes the spatial dispersion. Following w4,20x, the evolutionary equation for a local the order parameter is represented as:
E X Ž R ,t .
Et
q 12 bX 2
2
ž /5
d Ueff
sy
qD
ž
T y Tc Tc
E 2 E Ž R ,t . E R2
/
q bE Ž R ,t . E Ž R ,t .
.
Ž 23 .
The time evolution of EŽ R,t . is the ‘fast’ processes Žsee below.. For the stationary and homogeneous state the last equation accepts the form: 2
² X 2: q
T y TC TC b
² X 2 :s
x T2 b
,
E Ž R ,t . '² X 2 :.
Ž 24 .
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66
Using parameter ´ s X T2 b < 1, we shall write down the solution for three allocated states:
²X
2
°X
2 T
:s P~ X
2 T
TC
T ) TC
at
T y TC
² d X d X :r s
1r2
1
ž /
at
X T2 b
TC y T
¢Tb
T s TC
Ž 25 .
In the self-consistent approximation on the second moment it is possible to use the following kinetic equation:
Et
E s
EX
D
Ef EX
DŽ x .
q
2
²X :
2
Xf q D
E f E R 2.
.
y1r2
ž
exp y
X2 2² X 2 :
/
.
kB N 8 ´ TC
;
´ T C s X T2C b.
rC
n
ž /
exp y
T
,
rC2 s g
TC
,
.
T y TC
Ž 29 .
Thus correlation is defined by the spatial dispersion coefficient g and covers all volume of a sample. 4.1. Account of OZ formula on a basis of fluctuation–dissipation theorem (FDT) We shall carry out here follow stated in paragraph 101 in w20x.. The spatial-temporary spectral density d X is determined through an imaginary part of the response on FDT Žsee w21x.. In a result
Ž 27 .
The average and the most probable values of polarisation in this approximation coincide. The information on phase transition is contained in the dispersion ² X 2 :, which at all temperatures is defined by the solution of the Eq. Ž24.. The temperature dependence of the heat capacity is defined by the symmetric curve. The jump of the heat capacity is equal to zero. The height of peak is defined by the formula C Ž T y TC . s
²Ž d X . 2 :
Ž 26 .
The equilibrium solution of it is the Gaussian distribution: f Ž X . s Ž 2p² X 2 :.
4p rC2 r
r
C
T - TC
at
1
² Ž d X . 2 :s X T2 T yCT
C
Ef
the canonical Gibbs distribution with the effective GL functional. For the region T ) TC is presented in the form:
Ž 28 .
Height of the peak is of the same order, as size of jump in the Landau theory for one-domain states. So, for antiferroelectric the function C ŽT . represents symmetric, the peak – the jump is absent. See Fig. 5.18 in w19x. The experimental data in w19x are taken from work w20x..
X T2
2g
Ž d Xd X . v , k s
ž
v2q G
T y TC TC
2
q Dk 2
/
n
.
Ž 30 . Carrying out integration on frequency, we find, appropriate, the spatial spectral density. By the Fourier transformation is received the expression for simultaneous correlator the OZ-formula Ž29.. The given derivation leaves, however, without the answer two questions. First, the account leads results in ‘asymmetric’ expression for the spectral line Ž30. – the width of a line is defined by the sum two dissipative, that is not reflected in structure of the numerator. Secondly, for account of fluctuations it is possible to use the response or on external force, or on, appropriate, random Langevin source. However, both ways are equivalent only at some nonphysical choice of the Langevin source.
4. Fluctuations at phase transition. Ornstein– Zernike formula
4.2. LangeÕin source in REGL
The formula Ornstein–Zernike ŽOZ. for the simultaneous correlator can be established on basis of
Let’s designate the Langevin source through y Ž R,t . and writes down the connection of the expres-
Yu.L. KlimontoÕichr Physics Letters A 273 (2000) 61–69
sion for the spatial-temporary spectral densities Ž d X d X .v , k and Ž yy .v , k
Ž yy . v , k
Ž d Xd X . v , k s
ž
v2q G
T y TC TC
2
q Dk 2
.
Ž 31 .
/
Ž yy . v , k s 2 G
X T2 n
T ) TC the spectral density Ž yy .v , k is determined by expression
Ž yy . v , k s 2 Ž gŽ x . q Dk gŽ x . s
We see, that for the coincidence of the formulas Ž30., Ž31. the spectral density and the correlator of the Langevin source it is necessary to determine by the expression
67
T y TC v 02
.
²Ž d X . 2 : n
, T
² Ž d X . 2 :s X T2 T yCT
,
g
TC
2
.
Ž 33 .
C
In result is received the more realistic expression for the spectral density of fluctuations:
Ž d Xd X . v , k s ;
2 Ž g Ž x . q Dk 2 . 2
v q Ž gŽ x . q Dk
1
2 2
.
n
² Ž d X . 2 :. Ž 34 .
² y Ž R ,t . y Ž RX ,tX . :s 2 G
X T2 n
With its help we find the expressions for the spatial spectral density and the spatial correlation function
d Ž R y RX .
=d Ž t y tX . .
Ž 32 .
Thus, the Langevin source is d-correlated both on space, and on time. The intensity of noise is defined only the dissipative coefficient G and does not depend on the diffusion coefficient. Thus the given scheme of the account based on the FDT, corresponds to the Langevin source, in which is not taken into account a ‘diffusion noise’ is proportional Dk 2 . It puts under doubt, above mentioned the derivation of the OZ formula Ž29., as thus is not taken into account the contribution of a spatial diffusion. Its account radically changes behavior of fluctuations in region a critical point. Let’s carry out account fluctuations with the Langevin source, which is determined by the structure of two dissipative terms – of two ‘collision integrals’ in the kinetic equation for the distribution function f Ž X, R,t .. The intensity of the Langevin source in the kinetic equation for the distribution function f Ž X, R,t . is determined by the formula Ž19.6.4. in w8x or Ž8.5. in w9x. The appropriate moments of the Langevin source are determined by expression Ž19.6.14. in w8x or Ž9.2. in w9x. In this case the intensity of the Langevin source is determined by the sum of the dissipative ‘reaction’ and diffusion contributions. For the region
Ž d Xd X . k s
1
2 ² Ž d X . :; n
² d X d X :R , RX s
²Ž d X . 2 : n
d Ž R y RX . .
Ž 35 .
The spatial correlation now is different from zero only within the limit of volume of point Žphysically infinitesimal volume Vph ., as the function d Ž R y RX . y1 N Rs RX s Vph . For the one-dot correlator ² d X d X :RsRX we have, hence, the expression:
² d X d X :Rs RX s
1 Nph
² Ž d X . 2 :,
Nph s nVph .
Ž 36 .
3 Here Nph s nVph ' nl ph is a number of particles in a physically infinitesimal volume. The last formula shows, that the dispersion of fluctuation smoothed on the volume of continuous medium ‘point’, in Nph of time it is less, than on the Boltzmann distribution. The 0Z formula plays now other role. It is connected not to integral on spectral line on frequency v , but with a spectral line on zero frequency. Really,
Ž d X d X . vs0, kgŽ x . s
2
²Ž d X . 2 :
1 q rC2 k 2
n
.
Ž 37 .
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68
Let’s carry out in this expression the Fourier integration on wave numbers. In result receive expression
Ž d X d X . vs0, r gŽ x . s rC2 s X T2
TC
1 2p rC2 r
r
²Ž d X . 2 :
rC
n
ž /
exp y
.
T y TC
,
Ž 38 .
The right part of this expression is similar to the formula 0Z for the spatial correlator of fluctuations. 4.3. Fast fluctuations The time relaxation in the critical region is defined by the expression 1
tŽ E .
' DŽ E . Ž k . s 2 G
T y TC
ž
TC
/
q 2 b² X 2 : q Dk 2 .
Ž 39 . Average ‘energy’ ² X 2 :' ² E : is follow from the Eq. Ž24. for all temperatures. In the critical point Žat T s TC . we have:
t Ž E . Ž k ,T s TC . s
rC2 Ž T s TC . s
1
1
4 G Ž 1 q rC2 k 2 .
X T2
(
4 x T2 b
(´
These formulas are fair at all values of temperature. For temperatures is lower critical the relaxation time and radius of correlation we have: 1 1 t Ž X . Ž k ,T . s , y1 2 G Ž1qb k . ´ T rC2 Ž T . s
TC b
,
.
Ž 40 .
So, time of correlation and radius of correlation have the finite values in the critical point and at removal from the critical point fall down under the Curie law.
t Ž X . Ž k ,T . '
1 s
D Ž X ,T .
rC2 Ž T . s ² X 2 :.
² X 2:
G X T2 Ž 1 q rC2 k 2 .
2 D Ž X ,T .
² X 2:
v 2 q D 2 Ž X ,T .
n
Ž XX . k s
² X 2:
² XX :Ry RX s
n
Ž 43 .
² X 2: n ² X 2: Nph
d Ž R y RX . ; .
Ž 44 .
The coincidence with the appropriate formulas for ‘fast’ fluctuations has a place only for T ) TC . The OZ formula again is fair only for the correlator on zero frequency:
s
Ž 41 .
.
;
Ž d X d X . vs0, < RyRX < DŽ X . ² X 2: 1
,
Ž 42 .
T
The width of a line DŽ X . and the dispersion ² X 2 : are defined for all values of temperature. By integration on v we find the spatial spectral density and the spatial correlator:
² XX :Rs RX s
1
4 X T2 .
'
4.4. Slow fluctuations The time, the width of the spectral line and appropriate radius of correlation are defined by the formulas:
2 T
Thus, radius of correlation is defined by the coefficient 1rb . The radius of correlation is the order l ph and, thus, volume of sphere with radius rC corresponds to volume of a point of continuous medium. Similarly to behavior of radius of correlation a static susceptibility grows under the Curie law only at approach to a critical point from the party of symmetric phase. After the passages of the critical point it continues to grow and leaves on a saturation. It is possible to speak, thus, about presence of ‘jump of a susceptibility’. His size approximately same, as the Landau jump for the heat capacity. The expression for the spatial-temporary spectral density has form:
Ž XX . v , k s
TC
™ X´
TC y T
n
2p rC2 < R y RX <
ž
exp y
< R y RX < rC
/
.
Ž 45 .
Yu.L. KlimontoÕichr Physics Letters A 273 (2000) 61–69
For ‘slow’ fluctuations, radius of correlation is the minimal macroscopic characteristic. It means, that in an asymmetric phase interaction at temperatures T TC carries collective character. 4.5. The correlation parameter
tions. For slow fluctuations rC grows in process of downturn of temperature and comes nearer to finite value 1r 'b and, hence, remains macroscopic at all temperatures T - Tc . It gives the basis to determine physically infinitesimal small length as follows:
™ '1b ;
We can enter correlation parameter K – analogue of the Ginsburg number Gi for an asymmetric phase. The parameter K for all values of temperature is defined by the formula: 1 Ž d X d X . vs0, r C DŽ x . Ks ; 3 , rC2 s ² X 2 :. 2 ²X : nrC Ž 46 .
l ph s rC
It is small and in the critical point:
References
K Ž T s Tc . s
1 nrC3
s
Ž x T2 b . 3r4 nx T3
´ 3r4 s
nx T3
< 1.
Ž 47 . At the phase transition Žat T - TC . in process of downturn of temperature the correlation parameter monotonously decreases, that means reduction of a role of fluctuations and, hence, increase of a degree of coherence at the transition in an asymmetric phase. Thus, at downturn of temperature the domain of applicability of considered self-consistent approximation is improved. 4.6. Radius of correlation and size of a point of continuous medium The OZ formula on structure is similar to the Debye formula in the theory of plasma. The radius of correlation rC is defined by the Debye radius r D . The correlation parameter corresponds to the plasma parameter 1 1 Ks 3 ms 3 . Ž 48 . nrC nr D
m
Thus in a plasma the physically infinitesimal length l ph is of the order r D and, hence, the number of particles in a point is determined by the number of particles in sphere with the Debye radius, see w8x, section V.II, Nph ; Nr D3 s 1rm. We use this analogy for a definition of physically infinitesimal scales at phase transitions. For polidomain states two correlation radius were determined, accordingly, for fast and slow fluctua-
69
Nph s nrC3 s
n ™ K b 1
3r2
at
T
™ 0.
Ž 49 .
Radius of correlation defines also the width of a domain wall.
w1x L.D. Landau, JETP 7 Ž1937. 627. w2x L.D. Landau, E, M. Lifshits, Statistical Physics, Nauka, Moscow, 1976. w3x H.E. Stanley, Introduction to Phase Transition and Critical Phenomena, Clarendon Press, Oxford, 1971. w4x A.Z. Patashinskii, V.L. Pokrovskii, Fluctuation Theory of Phase transitions, Nauka, Moscow, 1982. w5x Sh. Ma, Modern Theory of Critical Phenomena, Benjamin, London, Mir, Moscow, 1980. w6x M.A. Anisimov, Critical Phenomena in Liquids and Liquid Crystals, Nauka, Moscow, 1987. w7x Yu.L. Klimontovich, Statistical Physics, Nauka, Moscow, 1982, Harwood Academic Publishers, New York, 1986. w8x Yu.L. Klimontovich, Statistical Theory of Open Systems, V.I, Yanus, Moscow, 1995; Kluwer Academic Publishers, Dordrecht, 1995; V.II, Yanus, Moscow 1999. w9x Yu.L. Klimontovich, TMP 96 Ž1993. 1. w10x D. Hilbert, Selected Works, vol. 1, Factorial, Moscow, 1998, p. 431. w11x V.L. Ginsburg, Uspechi Fiz. Nauka 38 Ž1949. 490. w12x B.A. Strukov, A.P. Levanuk, Physical Fundamentals of Ferroelectrics Phenomena in Crystals, Nauka-Fizmatlit, Moscow, 1995. w13x Yu.L. Klimontovich, Phys. Lett. A 210 Ž1996. 65. w14x Yu.L. Klimontovich, Int. J. Bifurc. Chaos 8 Ž1998. 661. w15x J-C Toledano, P. Toledano, The Landau Theory of Phase Transitions, World Scientific, Hong Kong, 1994. w16x S.A. Romanchuk, Pis’ma v ZhTP 16 Ž2000. 86. w17x B. Rellinghous, J. Raestner, T. Schneider, E.F. Wassermann, Phys. Rev. B 51 Ž1995. 2983. w18x N.M. Plakida, Hightemperature superconductivity. International Program of Education, Moscow, 1996. w19x M.E. Lines, A.M. Glass, Principle and Applications of Ferroelectrics and Related Materials, Clarendon Press, Oxford, 1977. w20x M. Amin, B.A. Strukov, Physics 12 Ž1970. 2035. w21x E.M. Lifshits, L.P. Pitaevskii, Physical kinetics, Nauka, Moscow 1979.
Physics Letters A 273 Ž2000. 61–69 www.elsevier.nlrlocaterpla
To thermodynamics and kinetics of second order phase transitions Yu.L. Klimontovich Department of Physics, M.V. LomonosoÕ Moscow State UniÕersity, Vorob’eÕy gory, 119899 Moscow, Russia Received 21 July 1997; received in revised form 10 April 2000; accepted 11 July 2000 Communicated by V.M. Agranovich
Abstract The accounts of thermodynamic and kinetic characteristics at second order phase transitions in ferroelectrics and antiferroelectrics for all temperature in critical region are carried out. Basis for account serves the kinetic equation for a local distribution function of values of a polarization vector Žthe order parameter.. The comparison of theoretical results with experimental date is considered. q 2000 Elsevier Science B.V. All rights reserved. Keywords: Second order phase transitions; Thermodynamics and kinetics; Self-consistent approximation for the first and the second moments; Critical region
1. Introduction In the ‘fluctuation theory of the second order phase transitions’ the critical point in the thermodynamic limit in mathematical sense is particular point. In this point the generalized static susceptibility, correlation radius of the simultaneous correlator of the order parameter fluctuations and correlation time of fluctuations tend to infinity. Such behavior of thermodynamic functions is a consequence of the thermodynamic limit transition w1–8x. Such behavior contradicts, however, to spirit both of thermodynamics and kinetics, which are based on model of continuous medium. In w7,8x for different levels of description the concrete definition of physically infinitesimal scales, the size of a point of continuous medium are given. E-mail address: [email protected] ŽY.L. Klimontovich..
That allows to enter natural small parameter – a ratio of the physically infinitesimal lengths l ph , determining the size of ‘point’ of continuous medium, to the characteristic size of system L.. The parameter l ph rL for a continuous medium approximation is always small. The expansion on parameter l ph rL was used for the Boltzmann gas and the Debye plasma w8,9x. As a result the generalized kinetic equations were constructed. The additional dissipative terms by the spatial diffusion of the corresponding distribution functions are defined. The unified description of kinetic and hydrodynamic processes without of the perturbation theory on Knudsen number is described. The principal difficulties of the Knudsen perturbation theory are set aside. The efficiency of expansion upon an parameter l ph rL is shown not only for Boltzmann gas, Debye plasma and the nonlinear Brownian motion, but for the theory of the second order phase transitions. The
0375-9601r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 0 0 . 0 0 4 7 0 - 9
62
Yu.L. KlimontoÕichr Physics Letters A 273 (2000) 61–69
possibility arises to avoid ‘the problems of infinity’ – the thermodynamic and fluctuation characteristics are finite at all temperatures in the critical region. Such results corresponds to mathematical definition of the role of ‘infinity’, which with limiting clearness by Hilbert was formulated in his paper AOn the infinityB w10x: G the mathematical literature, if one more attentiÕe by look at it, abounds with nonsenses and absurds, for which an infinity is guilty in the most cases.H G Our general conclusion is as follows: in the realized kind infinity does not meet anywhere.H For the greater clearness of description an example of the elementary model of ferroelectrics and antiferroelectrics w11–14x are using. The results compare with the Landau theory w1– 4,13,14x. It is justified by that already at this level the alternative approach allows to enter the certain improvements in the theory of phase transitions of the second order. Despite of development of the more general ‘the fluctuation theory of phase transitions’, the Landau theory remains by powerful tool for the description of phase transitions in ferroelectrics w3,11,15x.
2. Landau and Boltzmann distributions. Thermodynamic functions 2.1. The Landau distribution function The theory of phase transitions developed by Landau, is a phenomenological: the exact effective Hamilton function Heff Žh ,T ., connected with the Hamilton function of system w13x, is replaced by model expression w8x. The most probable value hm .p for the distribution function f Žh ,t . plays a role of the order parameter at phase transitions. The effective Hamilton function Heff Žh ,T . is the additive characteristic and consequently is proportional to number of particles of system N or, accordingly, of the volume of system V: Heff Ž h ,T . s Nh eff Ž h ,T . .
Ž 1.
Here is entered designations h eff Žh ,T . for the effective Hamilton function in the account on one particle.
Replacing the exact effective Hamilton function by model expression, let’s receive ‘the Landau distribution’ for values of the order parameter f L Ž h ,T . s exp
N Ž c Ž T . y h eff Ž h T . . k BT
,
Ž 2.
where c ŽT . is the free energy in account on one particle. In a case of one the order parameter Žsee Ž144.3. in w2x. we write the following expression: h eff Ž h ,T . s
a T y TC 2
TC
b h 2 q h 4 y h F0 , 4
Ž 3.
where F0 is an external force connected to the order parameter h. At a zero external field from a condition of a minimum of function Heff Žh ,T . we find the most probable values hm .p , values of the order parameter:
hm .p s 0 hm .p s
at
T G TC ;
a T? y T b
TC
at
T F TC .
Ž 4.
The appropriate values of a susceptibility Žlinear response. is defined by the well-known formulas. From these formulas follows, that the order parameter in the critical point is equal to zero. It is impossible, however, to confirm this, as a susceptibility in the critical point tends to infinity and, hence, the critical point is unattainable. In the Gaussian 2 approximation the dispersion ² Ž dh . : is proportional to the factor 1rN and in a thermodynamic limit, at T / TC is equal to zero. 2.2. The Boltzmann distribution function To give the description of fluctuations, we replace the Landau distribution by the Boltzmann one f B Ž h ,T . s exp
Ž c Ž T . y heff Ž h T . . k BT
.
Ž 5.
It differs from the Landau distribution by absence of the factor N in the exponent. The account of thermodynamic functions on a basis of the Boltzmann distribution allows to carry out the account of thermodynamic functions in the critical region at all
Yu.L. KlimontoÕichr Physics Letters A 273 (2000) 61–69
values of temperature. For a concrete accounts the elementary model of ferroelectrics uses below: Heff Ž X ,T . ' Nh eff Ž X ,T . , h eff s
m v 02 2
X2
T y TC TC
q 12 bX 2 .
Ž 6.
The variable X defines the vector of polarization: P s enX. The most probable value of displacement Xm .p will play a role of the order parameter. b is the dissipative coefficient of nonlinearity. 2.3. The heat capacity of equilibrium and nonequilibrium states At the account C a by the distribution function f Ž X,a,T . with h eff Ž X,T . we can submitted the heat capacity as the sum of two contributions w13x C a s yT
E 2F
sT
E Seff
E Fth
. Ž 7. ET ET ET Thus, the heat capacity it is represented as sum of entropy and thermal force. It is consequence that the effective Hamilton function depends on temperature. In the Landau theory the account of thermodynamic functions is based on use a effective Hamilton function at the most probable value of internal parameter. At temperatures below critical are two symmetric values of the order parameter. The minima are divided by a barrier, height of which depends on temperature and aspires to zero at approach to the critical point. At phase transitions in the Landau theory for an nonsymmetric state the heat capacity, strictly speaking, corresponds to nonequilibrium state. On to measure of approach to the critical point the condition of the quasistatic state is broken. In result, appears the necessity for the definition of the heat capacity for nonequilibrium states. Such definition is possible to enter as follows. The kinetic equation for a local distribution function f Ž X. R.t . contains temperature as parameter and enables to find nonequilibrium distribution of values of the polarization – the function f Ž X. R.t ., at various values of temperature. It allows to enter the concept of free energy at a nonequilibrium condition w7,8x: 2
q
F Ž t < T . s² Heff :t y NS Ž t . .
The average effective energy and the entropy for nonequilibrium conditions, corresponding to given value of the temperature, are defined through the solution of the kinetic equation. Using definitions of free energy of equilibrium and nonequilibrium states it is possible to enter appropriate the Lyapunov functional:
L F s F Ž t < T . y FŽ o. Ž T . f Ž X . R.t . dR s k B Tn ln f Ž X . R.t . dX G 0. fŽ0. Ž X . V
H
Ž 9. With the help of the kinetic equation is established the second inequality d LF d Ž F Ž t < T . y FŽ o . Ž T . . s F 0. Ž 10 . dt dt The existence of the Lyapunov functional testifies the stability of equilibrium state, which at all values of temperature is defined by the Boltzmann distribution. The quasistatical states are unstable. The time of relaxation essentially depends on affinity to the critical point. The definition of nonequilibrium free energy, including the dependence from temperature as the parameter, allows to enter and definition of the heat capacity of a nonequilibrium state E 2 F Ž t
Ž 8.
63
m v 02 Ž TC y T . 4
bTC2
2
at
T - TC .
Ž 12 .
Yu.L. KlimontoÕichr Physics Letters A 273 (2000) 61–69
64
From here follows, that the height of a potential barrier, dividing of two the most probable states, is proportional N and, hence, is infinite in the thermodynamic limit. The corresponding Kramers time tends to infinity. For Boltzmann distribution the depth of a potential hole is finite and is defined by the internal characteristic parameter of system ´ T s x T2 b. In the affinity to the critical point it is remained finite. Finiteness of the height of a barrier at any values of temperature is one of advantages of the Boltzmann distribution. In the Landau theory the heat capacity is defined by: C L Ž x m .p .,T . s Tc
E SL ET
sN
m v 02 2 bTc
'
1 kN
´T 2
.
ple, in magnetic alloys w17x and in high-temperature superconductors w18x.
3. Kinetic theory of the second order phase transitions In w8,13x proposed the following kinetic equation for the local distribution function f Ž X, R,t .:
E f Ž X , R ,t . Et E s
EX
DŽ x .
Ef EX
q
1 E h eff Ž X ,a f .
EX
mg
f qD
E R2
.
Ž 14 .
Ž 13 .
This expression defines the jump of the heat capacity. This result shows, that the heat capacity approach to the critical point from different levels. The critical point in framework of the Landau theory is unattainable. The Landau definitions of the entropy and the heat capacity do not coincide with results under the Gibbs theory. It gives the basis to speak about ‘the Landau thermodynamics’. The Landau results give only the part of the Gibbs heat capacity, which is defined by thermal force in the point X s X m .p w13x. From experimental data are known that two levels of the heat capacity Žbelow and above of the critical point. are connected by smooth curve with a finite height of a peak. The analytical account can be carried out only for small deviations from the most probable values. In this case it is possible to replace the Boltzmann distribution by the Gaussian one. The temperature dependence with in taken into account as the entropy, so and the thermal force at all values of temperature it is possible to obtain only on the basis of the Boltzmann distribution w13x. Results of numerical account at different values of the parameter ´ T are given on a Fig. 1 in w16x and w8x, section V.II. The height of the peak of the heat capacity temperature dependence is finite in the critical point at all values of temperature. A maximum of curves are displaced from a critical point in the side of low temperatures. Such behavior is observed, for exam-
E 2f
The kinetic equation contains two diffusion coefficients: of the spatial diffusion D s k B Trmg characterizes the resolve of nonhomogeneity in the position the center of dipoles; in the space of relative displacement of ions X. The second of them is defined by the formula DŽ x . s DrN for the Landau theory. For the Boltzmann distribution DŽ x . s D: Ds
k BT mg
s G X T2 ;
Gs
v 02 g
,
X T2 s
k BT m v 02
.
Ž 15 .
3.1. The self-consistent equation for the first moment From the kinetic equation follows the hooked chain of the equations for moments ² X n :. In the self-consistent approximation for the first moment the distribution function f Ž X , R ,t . s d Ž X y X Ž R ,t . . , ² X :s X Ž R ,t . .
Ž 16 . The equation for the function X Ž R,t . has a form:
E X Ž R ,t . Et
sy
v 02 T y Tc g
qD
Tc
q bX 2 Ž R ,t . X Ž R ,t .
E 2 X Ž R ,t . E R2
.
Ž 17 .
This equation has the structure of a reaction-diffusion equation. Such equations serve by a base in the
Yu.L. KlimontoÕichr Physics Letters A 273 (2000) 61–69
65
theory of autowaves and autosolitons, and synergetics. The similar equation is used and in the theory of phase transitions w1–8x. It is named ‘Relaxation equation Ginsburg–Landau’ ŽREGL..
The reaction-diffusion equation can be established independently from the definition of the Landau functional Heff . The right part of the reaction-diffusion equation can be presented in form of a functional gradient from
3.2. REGL in Landau theory of phase transitions
E X Ž R ,t .
Following the Landau theory, we shall generalize expression for effective Hamilton function with taken into account the spatial distribution of the order parameter. It is corresponds to introduction of the appropriate Landau functional w2–6x, see also w8x, section V.II. For considered model it can be presented as: Heff s nm v 02
H
½
1 2
g EX q 2
T y Tc
X2
Tc
ER
Et
sy
Ueff s
d Heff
nm v 02
d X Ž R ,t .
H
.
Ž 19 .
Here is entered a phenomenological dissipative coefficient g .
½
Et
qg g
E R2
.
Ž 20 .
l
This equation only by designations Žg v 02rg ; g G D . differs from the reaction-diffusion equation. Thus the functional Heff plays in the Landau theory the double role: the role of the nonequilibrium potential in the REGL; the role of the effective Hamilton functional in appropriate the Gibbs distribution for the equilibrium state. These two roles, in general, are inconsistent.
l
Tc
q 12 bX 2
2
ž /5 ER
s2 DyG
q bX 2 Ž R ,t . X Ž R ,t .
E 2 X Ž R ,t .
2
dR.
Ž 22 .
For a polidomain and an antiferroelectric structures average meaning of the polarization vector is equal zero. More effective in this case is the selfconsistent approximation on second moment. The equation for the function EŽ R,t . s ² X 2 : has the form:
Et Tc
g
3.3. The self-consistent approximation for the second moment
E E Ž R ,t .
T y Tc
Ž 21 .
Here D is the coefficient of spatial diffusion.
E X Ž R ,t .
s yg
v 02 X 2 T y Tc
2
Ž 18 .
.
The nonequilibrium functional is defined by expression
D EX
dR.
g
d X Ž R ,t .
q
Here n is average density of number of particles. The factor g characterizes the spatial dispersion. Following w4,20x, the evolutionary equation for a local the order parameter is represented as:
E X Ž R ,t .
Et
q 12 bX 2
2
ž /5
d Ueff
sy
qD
ž
T y Tc Tc
E 2 E Ž R ,t . E R2
/
q bE Ž R ,t . E Ž R ,t .
.
Ž 23 .
The time evolution of EŽ R,t . is the ‘fast’ processes Žsee below.. For the stationary and homogeneous state the last equation accepts the form: 2
² X 2: q
T y TC TC b
² X 2 :s
x T2 b
,
E Ž R ,t . '² X 2 :.
Ž 24 .
Yu.L. KlimontoÕichr Physics Letters A 273 (2000) 61–69
66
Using parameter ´ s X T2 b < 1, we shall write down the solution for three allocated states:
²X
2
°X
2 T
:s P~ X
2 T
TC
T ) TC
at
T y TC
² d X d X :r s
1r2
1
ž /
at
X T2 b
TC y T
¢Tb
T s TC
Ž 25 .
In the self-consistent approximation on the second moment it is possible to use the following kinetic equation:
Et
E s
EX
D
Ef EX
DŽ x .
q
2
²X :
2
Xf q D
E f E R 2.
.
y1r2
ž
exp y
X2 2² X 2 :
/
.
kB N 8 ´ TC
;
´ T C s X T2C b.
rC
n
ž /
exp y
T
,
rC2 s g
TC
,
.
T y TC
Ž 29 .
Thus correlation is defined by the spatial dispersion coefficient g and covers all volume of a sample. 4.1. Account of OZ formula on a basis of fluctuation–dissipation theorem (FDT) We shall carry out here follow stated in paragraph 101 in w20x.. The spatial-temporary spectral density d X is determined through an imaginary part of the response on FDT Žsee w21x.. In a result
Ž 27 .
The average and the most probable values of polarisation in this approximation coincide. The information on phase transition is contained in the dispersion ² X 2 :, which at all temperatures is defined by the solution of the Eq. Ž24.. The temperature dependence of the heat capacity is defined by the symmetric curve. The jump of the heat capacity is equal to zero. The height of peak is defined by the formula C Ž T y TC . s
²Ž d X . 2 :
Ž 26 .
The equilibrium solution of it is the Gaussian distribution: f Ž X . s Ž 2p² X 2 :.
4p rC2 r
r
C
T - TC
at
1
² Ž d X . 2 :s X T2 T yCT
C
Ef
the canonical Gibbs distribution with the effective GL functional. For the region T ) TC is presented in the form:
Ž 28 .
Height of the peak is of the same order, as size of jump in the Landau theory for one-domain states. So, for antiferroelectric the function C ŽT . represents symmetric, the peak – the jump is absent. See Fig. 5.18 in w19x. The experimental data in w19x are taken from work w20x..
X T2
2g
Ž d Xd X . v , k s
ž
v2q G
T y TC TC
2
q Dk 2
/
n
.
Ž 30 . Carrying out integration on frequency, we find, appropriate, the spatial spectral density. By the Fourier transformation is received the expression for simultaneous correlator the OZ-formula Ž29.. The given derivation leaves, however, without the answer two questions. First, the account leads results in ‘asymmetric’ expression for the spectral line Ž30. – the width of a line is defined by the sum two dissipative, that is not reflected in structure of the numerator. Secondly, for account of fluctuations it is possible to use the response or on external force, or on, appropriate, random Langevin source. However, both ways are equivalent only at some nonphysical choice of the Langevin source.
4. Fluctuations at phase transition. Ornstein– Zernike formula
4.2. LangeÕin source in REGL
The formula Ornstein–Zernike ŽOZ. for the simultaneous correlator can be established on basis of
Let’s designate the Langevin source through y Ž R,t . and writes down the connection of the expres-
Yu.L. KlimontoÕichr Physics Letters A 273 (2000) 61–69
sion for the spatial-temporary spectral densities Ž d X d X .v , k and Ž yy .v , k
Ž yy . v , k
Ž d Xd X . v , k s
ž
v2q G
T y TC TC
2
q Dk 2
.
Ž 31 .
/
Ž yy . v , k s 2 G
X T2 n
T ) TC the spectral density Ž yy .v , k is determined by expression
Ž yy . v , k s 2 Ž gŽ x . q Dk gŽ x . s
We see, that for the coincidence of the formulas Ž30., Ž31. the spectral density and the correlator of the Langevin source it is necessary to determine by the expression
67
T y TC v 02
.
²Ž d X . 2 : n
, T
² Ž d X . 2 :s X T2 T yCT
,
g
TC
2
.
Ž 33 .
C
In result is received the more realistic expression for the spectral density of fluctuations:
Ž d Xd X . v , k s ;
2 Ž g Ž x . q Dk 2 . 2
v q Ž gŽ x . q Dk
1
2 2
.
n
² Ž d X . 2 :. Ž 34 .
² y Ž R ,t . y Ž RX ,tX . :s 2 G
X T2 n
With its help we find the expressions for the spatial spectral density and the spatial correlation function
d Ž R y RX .
=d Ž t y tX . .
Ž 32 .
Thus, the Langevin source is d-correlated both on space, and on time. The intensity of noise is defined only the dissipative coefficient G and does not depend on the diffusion coefficient. Thus the given scheme of the account based on the FDT, corresponds to the Langevin source, in which is not taken into account a ‘diffusion noise’ is proportional Dk 2 . It puts under doubt, above mentioned the derivation of the OZ formula Ž29., as thus is not taken into account the contribution of a spatial diffusion. Its account radically changes behavior of fluctuations in region a critical point. Let’s carry out account fluctuations with the Langevin source, which is determined by the structure of two dissipative terms – of two ‘collision integrals’ in the kinetic equation for the distribution function f Ž X, R,t .. The intensity of the Langevin source in the kinetic equation for the distribution function f Ž X, R,t . is determined by the formula Ž19.6.4. in w8x or Ž8.5. in w9x. The appropriate moments of the Langevin source are determined by expression Ž19.6.14. in w8x or Ž9.2. in w9x. In this case the intensity of the Langevin source is determined by the sum of the dissipative ‘reaction’ and diffusion contributions. For the region
Ž d Xd X . k s
1
2 ² Ž d X . :; n
² d X d X :R , RX s
²Ž d X . 2 : n
d Ž R y RX . .
Ž 35 .
The spatial correlation now is different from zero only within the limit of volume of point Žphysically infinitesimal volume Vph ., as the function d Ž R y RX . y1 N Rs RX s Vph . For the one-dot correlator ² d X d X :RsRX we have, hence, the expression:
² d X d X :Rs RX s
1 Nph
² Ž d X . 2 :,
Nph s nVph .
Ž 36 .
3 Here Nph s nVph ' nl ph is a number of particles in a physically infinitesimal volume. The last formula shows, that the dispersion of fluctuation smoothed on the volume of continuous medium ‘point’, in Nph of time it is less, than on the Boltzmann distribution. The 0Z formula plays now other role. It is connected not to integral on spectral line on frequency v , but with a spectral line on zero frequency. Really,
Ž d X d X . vs0, kgŽ x . s
2
²Ž d X . 2 :
1 q rC2 k 2
n
.
Ž 37 .
Yu.L. KlimontoÕichr Physics Letters A 273 (2000) 61–69
68
Let’s carry out in this expression the Fourier integration on wave numbers. In result receive expression
Ž d X d X . vs0, r gŽ x . s rC2 s X T2
TC
1 2p rC2 r
r
²Ž d X . 2 :
rC
n
ž /
exp y
.
T y TC
,
Ž 38 .
The right part of this expression is similar to the formula 0Z for the spatial correlator of fluctuations. 4.3. Fast fluctuations The time relaxation in the critical region is defined by the expression 1
tŽ E .
' DŽ E . Ž k . s 2 G
T y TC
ž
TC
/
q 2 b² X 2 : q Dk 2 .
Ž 39 . Average ‘energy’ ² X 2 :' ² E : is follow from the Eq. Ž24. for all temperatures. In the critical point Žat T s TC . we have:
t Ž E . Ž k ,T s TC . s
rC2 Ž T s TC . s
1
1
4 G Ž 1 q rC2 k 2 .
X T2
(
4 x T2 b
(´
These formulas are fair at all values of temperature. For temperatures is lower critical the relaxation time and radius of correlation we have: 1 1 t Ž X . Ž k ,T . s , y1 2 G Ž1qb k . ´ T rC2 Ž T . s
TC b
,
.
Ž 40 .
So, time of correlation and radius of correlation have the finite values in the critical point and at removal from the critical point fall down under the Curie law.
t Ž X . Ž k ,T . '
1 s
D Ž X ,T .
rC2 Ž T . s ² X 2 :.
² X 2:
G X T2 Ž 1 q rC2 k 2 .
2 D Ž X ,T .
² X 2:
v 2 q D 2 Ž X ,T .
n
Ž XX . k s
² X 2:
² XX :Ry RX s
n
Ž 43 .
² X 2: n ² X 2: Nph
d Ž R y RX . ; .
Ž 44 .
The coincidence with the appropriate formulas for ‘fast’ fluctuations has a place only for T ) TC . The OZ formula again is fair only for the correlator on zero frequency:
s
Ž 41 .
.
;
Ž d X d X . vs0, < RyRX < DŽ X . ² X 2: 1
,
Ž 42 .
T
The width of a line DŽ X . and the dispersion ² X 2 : are defined for all values of temperature. By integration on v we find the spatial spectral density and the spatial correlator:
² XX :Rs RX s
1
4 X T2 .
'
4.4. Slow fluctuations The time, the width of the spectral line and appropriate radius of correlation are defined by the formulas:
2 T
Thus, radius of correlation is defined by the coefficient 1rb . The radius of correlation is the order l ph and, thus, volume of sphere with radius rC corresponds to volume of a point of continuous medium. Similarly to behavior of radius of correlation a static susceptibility grows under the Curie law only at approach to a critical point from the party of symmetric phase. After the passages of the critical point it continues to grow and leaves on a saturation. It is possible to speak, thus, about presence of ‘jump of a susceptibility’. His size approximately same, as the Landau jump for the heat capacity. The expression for the spatial-temporary spectral density has form:
Ž XX . v , k s
TC
™ X´
TC y T
n
2p rC2 < R y RX <
ž
exp y
< R y RX < rC
/
.
Ž 45 .
Yu.L. KlimontoÕichr Physics Letters A 273 (2000) 61–69
For ‘slow’ fluctuations, radius of correlation is the minimal macroscopic characteristic. It means, that in an asymmetric phase interaction at temperatures T TC carries collective character. 4.5. The correlation parameter
tions. For slow fluctuations rC grows in process of downturn of temperature and comes nearer to finite value 1r 'b and, hence, remains macroscopic at all temperatures T - Tc . It gives the basis to determine physically infinitesimal small length as follows:
™ '1b ;
We can enter correlation parameter K – analogue of the Ginsburg number Gi for an asymmetric phase. The parameter K for all values of temperature is defined by the formula: 1 Ž d X d X . vs0, r C DŽ x . Ks ; 3 , rC2 s ² X 2 :. 2 ²X : nrC Ž 46 .
l ph s rC
It is small and in the critical point:
References
K Ž T s Tc . s
1 nrC3
s
Ž x T2 b . 3r4 nx T3
´ 3r4 s
nx T3
< 1.
Ž 47 . At the phase transition Žat T - TC . in process of downturn of temperature the correlation parameter monotonously decreases, that means reduction of a role of fluctuations and, hence, increase of a degree of coherence at the transition in an asymmetric phase. Thus, at downturn of temperature the domain of applicability of considered self-consistent approximation is improved. 4.6. Radius of correlation and size of a point of continuous medium The OZ formula on structure is similar to the Debye formula in the theory of plasma. The radius of correlation rC is defined by the Debye radius r D . The correlation parameter corresponds to the plasma parameter 1 1 Ks 3 ms 3 . Ž 48 . nrC nr D
m
Thus in a plasma the physically infinitesimal length l ph is of the order r D and, hence, the number of particles in a point is determined by the number of particles in sphere with the Debye radius, see w8x, section V.II, Nph ; Nr D3 s 1rm. We use this analogy for a definition of physically infinitesimal scales at phase transitions. For polidomain states two correlation radius were determined, accordingly, for fast and slow fluctua-
69
Nph s nrC3 s
n ™ K b 1
3r2
at
T
™ 0.
Ž 49 .
Radius of correlation defines also the width of a domain wall.
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