Statistical Mechanics by Konstantin K. Likharev - HTML preview
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w U T w ,
0
i.e.
.
(5.123)
w
T
Since the left-hand side of the last relation is just (ln w), it may be easily integrated over q, giving U
U (q)
ln w
ln C,
w
i.e. (r) C exp
,
(5.124)
T
T
where C is a normalization constant. With both sides multiplied by the number N of similar, independent systems, with the spatial density n(q) = Nw(q), this equality becomes the Boltzmann distribution (3.26).
As a less trivial example of the Smoluchowski equation’s applications, let us use it to solve the 1D Kramers problem (Fig. 10) in the corresponding high-damping limit, m << A, where A (still to be calculated) is some time scale of the particle’s motion inside the well. It is straightforward to verify that the 1D version of Eq. (121),
1 U
w
I
,
(5.125a)
w
w
T
q
q
(where Iw is the probability current at point q, rather than its density) is mathematically equivalent to T
U ( q)
U ( q)
I exp
exp
,
(5.125b)
w
w
T q
T
so we may write
U ( q)
T
U ( q)
I exp
exp
.
(5.126)
w
w
T
q
T
66 It is named after Marian Smoluchowski, who developed this formalism in 1906. Note that sometimes Eq. (122) is referred to as the Fokker-Planck equation, but it is more common to use that name for a more general equation discussed in the next section.
Chapter 5
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As was discussed above, the notion of a metastable state’s lifetime is well-defined only for sufficiently low temperatures
T U .
(5.127)
0
when the lifetime is relatively long: >> A. Since according to Eq. (111a), the first term of the continuity equation (117b) has to be of the order of W/, in this limit the term, and hence the gradient of Iw, are exponentially small, so the probability current virtually does not depend on q in the potential barrier region. Let us use this fact in the integration of both sides of Eq. (126) over that region: q"
U
( q)
T
U
( q q"
I
)
exp
dq
w exp
,
(5.128)
w
T
T
q'
q'
where the integration limits q’ and q” (shown schematically in Fig. 10) are selected so that T U ( q' ) U ( q ), U ( q ) U ( q" ) U .
(5.129)
1
2
0
(Obviously, such selection is only possible if condition (127) is satisfied.) In this limit, the contribution from the point q” to the right-hand side of Eq. (129) is negligible because the probability density behind the barrier is exponentially small. On the other hand, the probability at the point q’ has to be close to the value given by its quasi-stationary Boltzmann distribution (124), so
U ( q' )
U ( q )
1
w( q' ) exp
w( q )exp
,
(5.130)
1
T
T
and Eq. (128) yields
q"
T
U ( q) U ( q )
1
I
w( q )
exp
.
(5.131)
w
1
dq
q'
T
Patience, my reader, we are almost done. The probability density w( q 1) at the well’s bottom may be expressed in terms of the total probability W of the particle being in the well by using the normalization condition
W
U ( q ) U ( q)
1
(
w q ) exp
dq ;
(5.132)
1
T
s
well'
bottom
the integration here may be limited to the region where the difference U( q) – U( q 1) is much smaller than U 0 – cf. Eq. (129). According to the Taylor expansion, the shape of virtually any smooth potential U( q) near the point q 1 of its minimum may be well approximated with a quadratic parabola: 2
d U
U ( q q ) U ( q )
1
( q q )2 ,
where
0.
(5.133)
1
1
2
1
1
2
q q 1
dq
With this approximation, Eq. (132) is reduced to the standard Gaussian integral:67
2
~
1/ 2
( q q )
2
q
2 T
1
1
1
~
W (
w q )
exp
dq (
w q ) exp
q
d (
w q )
.
(5.134)
1
2
1
T
2
1
T
s
well'
1
bottom
67 If necessary, see MA Eq. (6.9b) again.
Chapter 5
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To complete the calculation, we may use a similar approximation for the barrier top:
U ( q q ) U ( q ) U ( q ) 2
( q q )2 U ( q )
2
U
( q q )2 ,
2
1
2
2
2
1
0
2
2
(5.135)
2
d U
where
,
0
2
2
q q 2
dq
and work out the remaining integral in Eq. (131), because in the limit (129) it is dominated by the contribution from a region very close to the barrier top, where the approximation (135) is asymptotically exact. As a result, we get
q"
1/ 2
U ( q) U ( q )
U
2 T
1
0
exp
dq exp
.
(5.136)
q'
T
T
2
Plugging Eq. (136), and the w( q 1) expressed from Eq. (134), into Eq. (131), we finally get 1/ 2
( )
1 2
U 0
I W
exp
.
(5.137)
w
2
T
This expression should be compared with the 1D version of Eq. (117b) for the segment [–, q’].
Since this interval covers the region near q 1 where most of the probability density resides, and Iq(-) =
0, this equation is merely
dW
I ( q' ) 0 .
(5.138)
dt
w
In our approximation, Iw( q’) does not depend on the exact position of the point q’, and is given by Eq.
(137), so plugging it into Eq. (138), we recover the exponential decay law (111a), with the lifetime
obeying the Arrhenius law (111b), and the following attempt time:
Kramers
2
formula
2
,
where
.
(5.139)
A
for high
1/ 2
1 2 1/ 2
damping
1 2
,
1 2
,
1 2
Thus the metastable state lifetime is indeed described by the Arrhenius law, with the attempt time scaling as the geometric mean of the system’s “relaxation times” near the potential well bottom (1) and the potential barrier top (2).68 Let me leave it for the reader’s exercise to prove that if the potential profile near the well’s bottom and/or top is sharp, the expression for the attempt time should be modified, but the Arrhenius decay law (111) is not affected.
5.7. The Fokker-Planck equation
Formula (139) is just a particular, high-damping limit of a more general result obtained by Kramers. In order to get all of it (and much more), we need to generalize the Smoluchowski equation to arbitrary values of damping . In this case, the probability density w is a function of not only the particle’s position q (and time t) but also of its momentum p – see Eq. (2.11). Thus the continuity equation (117) needs to be generalized to the 6D phase space {q, p}. Such generalization is very natural: 68 Actually, 2 describes the characteristic time of the exponential growth of small deviations from the unstable fixed point q 2 at the barrier top, rather than their decay, as near the stable point q 1.
Chapter 5
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w
j j 0,
(5.140)
q
q
p
p
t
where j q (which was called j w in the last section) is the probability current density in the coordinate space, and q (which was denoted as in that section) is the usual vector operator in the space, while j p
is the current density in the momentum space, and p is the similar vector operator in that space: 3
3
n
n
.
(5.141)
q
j
p
j
j1
q
p
j
j1
j
At negligible fluctuations ( T 0), j p may be composed using the natural analogy with j q – see Eq. (119). In our new notation, that relation reads,
p
j wq w ,
(5.142)
q
m
so it is natural to take
j wp w
,
(5.143a)
p
F
where the (statistical-ensemble) averaged force F includes not only the contribution due to the potential’s gradient but also the drag force –v provided by the environment – see Eq. (64) and its discussion:
p
j w( U v) w( U ) .
(5.143b)
p
q
q
m
As a sanity check, it is straightforward to verify that the diffusion-free equation resulting from the combination of Eqs. (140), (142) and (143),
w
p
p
,
(5.144)
drift
q
w
p
w U
q
t
m
m
allows the following particular solution:
w q
( , p, t) q q t p p t, (5.145)
where the statistical-averaged coordinate and momentum satisfy the deterministic equations of motion, p
p
q
,
p U
,
(5.146)
m
q
m
describing the particle’s drift, with the usual deterministic initial conditions.
In order to understand how the diffusion should be accounted for, let us consider a statistical ensemble of free ( qU = 0, = 0) particles that are uniformly distributed in the direct space q (so qw =
0), but possibly localized in the momentum space. In this case, the right-hand side of Eq. (144) vanishes, i.e. the time evolution of the probability density w may be only due to diffusion. In the corresponding limit F 0, the Langevin equation (107) for each Cartesian coordinate is reduced to
~
~
q
m F ( t),
i.e. p F ( t) .
(5.147)
j
j
j
j
Chapter 5
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The last equation is identical to the high-damping 1D equation (74) (with Fdet = 0), with the replacement q pj/, and hence the corresponding contribution to w/ t may be described by the last term of Eq.
(122), with that replacement:
w
2
T
D
w
2
2
w T 2
w
t diffusion
p /
.
(5.148)
p
p
Now the reasonable assumption that in the arbitrary case, the drift and diffusion contributions to w/ t just add up immediately leads us to the full Fokker-Planck equation:69
Fokker-
w
p
Planck
w w U p T 2
w .
(5.149)
equation
t
q
m
p
q
m
p
As a sanity check, let us use this equation to calculate the stationary probability distribution of the momentum of particles with an arbitrary damping but otherwise free, in the momentum space, assuming (just for simplicity) their uniform distribution in the direct space, q = 0. In this case, Eq.
(149) is reduced to
p
p
2
.
(5.150)
p
w
T w ,
0
i.e.
p
p
w T
w
p
0
m
m
The first integration over the momentum space yields
p
p 2
w T w j ,
i.e. w
T w j ,
(5.151)
m
p
w
p 2 m
p
w
where j w is a vector constant describing a possible general probability flow in the system. In the absence of such flow, j w = 0, we get
p 2
w
2
2
p
p
p
T
T ln w ,
0
giving w const exp
,
(5.152)
p
p
2 m
w
2 m
2 mT
i.e. the Maxwell distribution (3.5). However, the result (152) is more general than that obtained in Sec.
3.1, because it shows that the distribution stays the same even at non-zero damping. It is also straightforward to verify that in the more general case of an arbitrary stationary potential U(q), Eq. (149) is satisfied with the stationary solution (3.24), also giving j w = 0.
In the limit where the damping is large, i.e. the inertial effects are relatively small, the solution of the Fokker-Planck equation tends, relatively rapidly, to the following product
p p q t
0 , 2
(
w q,p, t) exp
w(q ,t) ,
(5.153)
2 mT
where p0 –( m/) qU, followed by a much slower time evolution of the direct-space distribution w(q, t), described by the Smoluchowski equation (122).
Another important particular case is that of a quasi-periodic motion of a particle, with low damping, in a soft potential well. In this case, the Fokker-Planck equation describes both the diffusion of 69 It was first derived by Adriaan Fokker in 1913 in his PhD thesis and further elaborated by Max Planck in 1917.
(Curiously, A. Fokker is more famous for his work on music theory, and the invention and construction of several new keyboard instruments, than for this and several other important contributions to theoretical physics.) Chapter 5
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the effective phase of such (generally nonlinear, “anharmonic”) oscillator, and a slow relaxation of its energy. If we are only interested in the latter effect, Eq. (149) may be reduced to the so-called energy diffusion equation,70 which is much easier to solve.
However, in most practically interesting cases, solutions of Eq. (149) are rather complicated.
(Indeed, the reader should remember that these solutions embody, in the particular case T = 0, all classical dynamics of a particle.) Because of this, I will present (rather than derive) only one more of them: the Kramers’ solution71 of his problem (Fig. 10) for / m2 ~ 1. In this general case, the metastable state’s lifetime turns out to be again given by the Arrhenius formula (111b), with the following reciprocal attempt time:
1/ 2
2
1
η
m
1
2
/ 2 ,
for
/
,
1
1
2
(5.154)
2
2
2
m
m
A
2
4
2
1/ 2
m / 2 1/ 2
η
m
1
2
1 2 ,
1
for /
,
2
where 1,2 (1,2/ m)1/2. Thus, in the limit / m2 << 1, Eqs. (111b) and (154) give a very simple result 2
U 0
exp
.
(5.155)
1
T
Note, however, that this result is strictly valid only if / m2 >> T/ U 0 (as a reminder, the latter ratio has to be much smaller than 1 in order for the very notion of lifetime to be meaningful) and at lower damping, its pre-exponential factor requires a correction, which may be calculated using the already mentioned energy diffusion equation.72
The Kramers’ result for the classical thermal activation of a system over a potential barrier may be compared with that for its quantum-mechanical tunneling through the barrier.73 The WKB
approximation for the latter effect gives the expression
2
2
( q)
exp
,
(5.156)
Q
A
2
( q) dq
,
with
U ( q) E
2 m
2
( )0
q
showing that generally, the classical and quantum lifetimes of a metastable state have different dependences on the barrier shape. For example, for a nearly rectangular potential barrier, the exponent that determines the classical lifetime (155) depends (linearly) only on the barrier height U 0, while that defining the quantum lifetime (156) is proportional to the barrier width and to the square root of U 0.
However, in the important case of “soft” potential profiles, which are typical for the case of emerging (or nearly disappearing) quantum wells (Fig. 11), the classical and quantum results are closely related.
U ( q)
q
1
U 0
q
q
0
2
Fig. 5.11. Cubic-parabolic potential
profile and its parameters.
70 An example of such an equation, for the particular case of a harmonic oscillator, is given by QM Eq. (7.214).
The Fokker-Planck equation, of course, can give only its classical limit, with n, n e >> 1.
71 H. Kramers, Physica 7, 284 (1940); see also the model solution of Problem 27.
72 See, e.g., the review paper by O. Mel’nikov, Phys. Repts. 209, 1 (1991), and references therein.
73 See, e.g., QM Secs. 2.4-2.6.
Chapter 5
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Indeed, such potential profile U( q) may be well approximated by four leading terms of its Taylor expansion, with the highest term proportional to ( q – q 0)3, near any point q 0 in the vicinity of the well. In this approximation, the second derivative d 2 U/ dq 2 vanishes at the inflection point q 0 = ( q 1 + q 2)/2, exactly between the well’s bottom and the barrier’s top (in Fig. 11, q 1 and q 2). Selecting the origin at this point, as this is done in Fig. 11, we may reduce the approximation to just two terms:74
b 3
U ( q) aq q .
(5.157)
3
(For the particle’s escape into the positive direction of the q-axis, we should have a, b > 0.) An easy calculation gives all essential parameters of this cubic parabola: the positions of its minimum and maximum:
q q a b
(5.158)
2
1
/ 1/2,
the barrier height over the well’s bottom:
1/ 2
3
4 a
U U ( q ) U ( q )
,
(5.159)
0
2
1
3
b
and the effective spring constants at these points:
2
d U
2 ab
.
(5.160)
1
2
1/2
2
dq q ,12
Hence for this potential profile, Eq. (155) may be rewritten as
Soft well:
2
U
(
2
)
0
2
ab 1/ 2
thermal
exp
,
with
.
(5.161)
lifetime
T
0
m
0
On the other hand, for the same profile, the WKB approximation (156) (which is accurate when the height of the metastable state energy over the well’s bottom, E – U( q 1) 0/2, is much lower than the barrier height U 0) yields75
Soft well:
2
1/2
36 U
quantum
0
0
exp
.
(5.162)
Q
lifetime
864 U
5
0
0
0
The comparison of the dominating, exponential factors in these two results shows that the thermal activation yields a lower lifetime (i.e., dominates the metastable state decay) if the temperature is above the crossover value
36
T
2
.
7
.
(5.163)
c
0
0
5
This expression for the cubic-parabolic barrier may be compared with a similar crossover for a quadratic-parabolic barrier,76 for which T c = 2 0 6.28 0. We see that the numerical factors for 74 As a reminder, a similar approximation arises for the P( V) function, at the analysis of the van der Waals model near the critical temperature – see Problem 4.6.
75 The main, exponential factor in this result may be obtained simply by ignoring the difference between E and U( q 1), but the correct calculation of the pre-exponential factor requires taking this difference, 0/2, into account
– see, e.g., the model solution of QM Problem 2.43.
76 See, e.g., QM Sec. 2.4.
Chapter 5
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the quantum-to-classical crossover temperature for these two different soft potential profiles are close to each other – and much larger than 1, which could result from a naïve estimate.
5.8. Back to the correlation function
Unfortunately, I will not have time/space to either derive or even review solutions of other problems using the Smoluchowski and Fokker-Planck equations, but have to mention one conceptual issue. Since it is intuitively clear that the solution w(q, p, t) of the Fokker-Planck equation for a system provides full statistical information about it, one may wonder how it may be used to find its temporal characteristics that were discussed in Secs. 4-5 using the Langevin formalism. For any statistical average of a function taken at the same time instant, the answer is clear – cf. Eq. (2.11): f q t
( ),p t
( ) f (q,p) w(q,p, t d 3
) qd 3 p ,
(5.164)
but what if the function depends on variables taken at different times, for example as in the correlation function Kf() defined by Eq. (48)?
To answer this question, let us start from the discrete-variable case when Eq. (164) takes the form (2.7), which, for our current purposes, may be rewritten as
f t f W t() .
(5.165)
m
m
m
In plain English, this is a sum of all possible values of the function, each multiplied by its probability as a function of time. But this implies that the average f( t) f( t’) may be calculated as the sum of all possible products fmfm’, multiplied by the joint probability to measure outcome m at moment t, and outcome m’ at moment t’. The joint probability may be represented as a product of Wm( t) by the conditional probability W( m’, t’ m, t). Since the correlation function is well defined only for stationary systems, in the last expression we may take t = 0, i.e. look for the conditional probability as the solution, Wm’(), of the equation describing the system’s probability evolution, at time = t’ – t (rather than t’), with the special initial condition
W (0)
.
(5.166)
m'
m' , m
On the other hand, since the average f( t) f( t +) of a stationary process should not depend on t, instead of Wm(0) we may take the stationary probability distribution Wm(), independent of the initial conditions, which may be found as the same special solution, but at time . As a result, we get Correlation
f t f t f W f W
.
(5.167) function:
m
m
m' m'
discrete
m, m'
system
This expression looks simple, but note that this recipe requires solving the time evolution equations for each Wm’() for all possible initial conditions (166). To see how this recipe works in practice, let us revisit the simplest two-level system (see, e.g., Fig. 4.13, which is reproduced in Fig. 12
below in a notation more convenient for our current purposes), and calculate the correlation function of its energy fluctuations.
Chapter 5
Page 36 of 44
SM: Statistical Mechanics
W ( t)
E
1
1
Fig. 5.12. Dynamics of a two-level system.
W 0 t
E 0
0
The stationary probabilities of the system’s states (i.e. their probabilities for ) have been calculated in problems of Chapter 2, and then again in Sec. 4.4 – see Eq. (4.68). In our current notation (Fig. 12),
W
W
0
1
1
1
,
,
/ T
/
T
1 e
e
1
(5.168)
that
so
E W W
0
0
1
.
/ T
e
1
To calculate the conditional probabilities Wm’( ) with the initial conditions (166) (according to Eq.
(168), we need all four of them, for { m, m’} = {0, 1}), we may use the master equations (4.100), in our current notation reading
dW
dW
1
0
W W .
(5.169)
0
1
d
d
Since Eq. (169) conserves the total probability, W 0 + W 1 = 1, only one probability (say, W 1) is an independent variable, and for it, Eq. (169) gives a simple, linear differential equation dW
1 1 W
W
W
,
(5.170)
1
,
where
1
1
d
which may be readily integrated for an arbitrary initial condition:
W
( ) W ( )
0 e
W ()
,
(5.171)
1
1
1
1 e
where W 1() is given by the second of Eqs. (168). (It is straightforward to verify that the solution for W 0() may be represented in a form similar to Eq. (171), with the corresponding replacement of the state index.)
Now everything is ready to calculate the average E( t) E( t +) using Eq. (167), with fm,m’ = E 0,1.
Thanks to our (smart :-) choice of the energy reference, of the four terms in the double sum (167), all three terms that include at least one factor E 0 = 0 vanish, and we have only one term left to calculate: 2
E( t) E( t ) E W () E W ( )
E W () W 0
e
W () 1
e
1
1
1
1
W (0) 1
1
1
1
1
W (0) 1
1
1
2
1
2
T
(5.172)
T
e
1
e
T
e
e
2 1
/
/
e
1
/
e
1
/
e T
.
1
From here and the last of Eqs. (168), the correlation function of energy fluctuations is77
77 The step from the first line of Eq. (173) to its second line utilizes the fact that our system is stationary, so E( t +
) = E( t) = E() = const.
Chapter 5
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SM: Statistical Mechanics
~
~
K ( ) E( t) E( t )
E
E( t) E( t) E( t ) E( t)
/
e T
(5.173)
E( t) E( t ) E 2
2
e
e T
,
/
2
1
so its variance, equal to KE(0), does not depend on the transition rates and . However, since the rates have to obey the detailed balance relation (4.103), / = exp{/ T}, for this variance we may formally write
KE 0
/ T
e
/
,
(5.174)
2
/ T
e
2
1
/
2
1
2
2
so Eq. (173) may be represented in a simpler form:
Energy
2
K
( )
e
.
(5.175) fluctuations:
E
2
two-level
system
We see that the correlation function of energy fluctuations decays exponentially with time, with the rate
. Now using the Wiener-Khinchin theorem (58) to calculate its spectral density, we get
2
1
Γ Γ
Γ
Δ
Γ Γ
2
S (
)
Δ
e
cos
.
(5.176)
E
d
2
2
2
Γ
Γ Γ
0
Such Lorentzian dependence on frequency is very typical for discrete-state systems described by master equations. It is interesting that the most widely accepted explanation of the 1/ f noise (also called the “flicker” or “excess” noise), which was mentioned in Sec. 5, is that it is a result of thermally-activated jumps between states of two-level systems with an exponentially-broad statistical distribution of the transition rates . Such a broad distribution follows from the Kramers formula (155), which is approximately valid for the lifetimes of both states of systems with double-well potential profiles (Fig.
13), for a statistical ensemble with a smooth statistical distribution of the energy barrier heights U 0.
Such profiles are typical, in particular, for electrons in disordered (amorphous) solid-state materials, which indeed feature high 1/ f noise.
U
U 0
Fig. 5.13. Typical double-
well potential profile.
0
q
Returning to the Fokker-Planck equation, we may use the following evident generalization of Eq. (167) to the continuous-variable case:
Correlation
f ( t) f ( t )
3
3
3
3
d qd p d q'd p' f (q , p)
w q , p, f (q ', p ' )
w q ', p ' ,
,
(5.177) function:
continuous
system
were both probability densities are particular values of the equation’s solution with the delta-functional initial condition
Chapter 5
Page 38 of 44
SM: Statistical Mechanics
(
w q ' , p ' )
0
, (q ' - q) (p ' - p) .
(5.178)
For the Smoluchowski equation, valid in the high-damping limit, the expressions are similar, albeit with a lower dimensionality:
f ( t) f ( t )
3
3
d q d q' f (q)
w q, f (q ' )
w q ' ,
,
(5.179)
(
w q ' )
0
, (q ' - q) .
(5.180)
To see this formalism in action, let us use it to calculate the correlation function Kq() of a linear relaxator, i.e. an overdamped 1D harmonic oscillator with m0 << . In this limit, as Eq. (65) shows, the oscillator’s coordinate averaged over the ensemble of environments obeys a linear equation,
q q 0 ,
(5.181)
which describes its exponential relaxation from the initial position q 0 to the equilibrium position q = 0, with the reciprocal time constant = /:
t
q t
( ) q e
.
(5.182)
0
The deterministic equation (181) corresponds to the quadratic potential energy U( q) = q 2/2, so the 1D version of the corresponding Smoluchowski equation (122) takes the form 2
w
w
wq T
.
(5.183)
2
t
q
q
It is straightforward to check, by substitution, that this equation, rewritten for the function w( q’,), with the 1D version of the delta-functional initial condition (180), w( q’,0) = ( q’ – q), is satisfied with a Gaussian function:
2
1
q' q ( )
(
w q' , )
,
(5.184)
2
exp
1/ 2
2
q( )
2 q ( )
with its center q() moving in accordance with Eq. (182), and a time-dependent variance 2
2
2
2
2
T
q
( ) q ()
1 e
,
where q () q
.
(5.185)
(As a sanity check, the last equality coincides with the equipartition theorem’s result.) Finally, the first probability under the integral in Eq. (179) may be found from Eq. (184) in the limit (in which
q() 0), by replacing q’ with q:
1
2
q
(
w q,)
(5.186)
2
exp
.
1/ 2 q
()
2 2
q
()
Now all ingredients of the recipe (179) are ready, and we can spell it out, for f ( q) = q, as
2
1
2
q
q'
qe
q( t) q( t )
dq dq' q exp
q'
.
(5.187)
2
exp
2
2 q( ) q()
2 q ( )
2 q ( )
Chapter 5
Page 39 of 44
SM: Statistical Mechanics
The integral over q’ may be worked out first, by replacing this integration variable with ( q” + qe-) and hence dq’ with dq” :
1
2
2
q
q"
q( t) q( t )
q exp
dq q" qe
dq" (5.188)
2
exp
.
2 q( ) q
()
2 q
()
2 2
q
( )
The internal integral of the first term in the parentheses equals zero (as that of an odd function in symmetric integration limits), while that with the second term is the standard Gaussian integral, so: 1
q 2
2
2
T
q t
( ) q t
(
)
e
q exp
dq
e
2
exp 2
d . (5.189)
2
1/ 2 q()
2 q 2 ()
1/2
The last integral78 equals 1/2/2, so taking into account that for this stationary system centered at the coordinate origin, q() = 0, we finally get a very simple result: Correlation
T
~ ~
K ( ) q( t) q( t ) q( t) q( t ) q (5.190) function:
q
2 q( t) q( t )
e
.
linear
relaxator
As a sanity check, for = 0, it yields Kq(0) q 2 = T/, in accordance with Eq. (185). As is increased the correlation function decreases monotonically – see the solid-line sketch in Fig. 8.
So, the solution of this very simple problem has required straightforward but somewhat bulky calculations. On the other hand, the same result may be obtained literally in one line using the Langevin formalism – namely, as the Fourier transform (59) of the spectral density (68) in the corresponding limit m << , with S () given by Eq. (73a):79
F
T
1
T
cos
T
K ( ) 2 S () cos d 2
cos d 2
d
(5.191)
q
q
e
2
2
( )
0
0
.
2
2
0
This example illustrates the fact that for fluctuations in linear systems (and small fluctuations in nonlinear systems) the Langevin approach is usually much simpler than the one based on the Fokker-Planck or Smoluchowski equations. However, again, the latter approach is indispensable for the analysis of fluctuations of arbitrary intensity in nonlinear systems.
To conclude this chapter, I have to emphasize again that the Fokker-Planck and Smoluchowski equations give a quantitative description of the time evolution of nonlinear Brownian systems with dissipation in the classical limit. The description of the corresponding properties of such dissipative (“open”) and nonlinear quantum systems is more complex,80 and only a few simple problems of their theory have been solved analytically so far,81 typically using particular models of the environment, e.g., as a large set of harmonic oscillators with various statistical distributions of their parameters, each leading to a specific function () for the generalized susceptibility.
78 See, e.g., MA Eq. (6.9c).
79 The involved table integral may be found, e.g., in MA Eq. (6.11).
80 See, e.g., QM Sec. 7.6.
81 See, e.g., the solutions of the 1D Kramers problem for quantum systems with low damping by A. Caldeira and A. Leggett, Phys. Rev. Lett. 46, 211 (1981), and with high damping by A. Larkin and Yu. Ovchinnikov, JETP
Lett. 37, 382 (1983).
Chapter 5
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5.10. Exercise problems
5.1. By treating the first 30 digits of number = 3.1415 as a statistical ensemble of integers k (equal to 3, 1, 4, 1, 5,…), calculate the average k and the r.m.s. fluctuation k. Compare the results with those for the ensemble of randomly selected decimal integers 0, 1, 2, 9.
5.2. An ideal classical gas of N similar particles fills a spherical cavity of radius R. Calculate the variance of fluctuations of the position r of its center of mass, in equilibrium.
5.3. Calculate the variance of fluctuations of a magnetic moment m placed into an external magnetic field H, within the same two models as in Problem 2.4:
(i) a quantum spin-½ with a gyromagnetic ratio , and
(ii) a classical magnetic moment m of a fixed magnitude m0 but an arbitrary orientation, both in thermal equilibrium at temperature T. Compare the results.82
Hint: Mind all three Cartesian components of the vector m.
5.4. For a field-free two-site Ising system with energy values Em = – Js 1 s 2, in thermal equilibrium at temperature T, calculate the variance of energy fluctuations. Explore the low-temperature and high-temperature limits of the result.
5.5. In a system in thermodynamic equilibrium with fixed T and , both the number N of particles and the internal energy E may fluctuate. Express the mutual correlation factor of these fluctuations via the average of E. Spell out the result for an ideal classical gas of N >> 1 particles.
5.6. As was mentioned in Sec. 2, the variance of energy fluctuations in a system with fixed T and
(i.e. a member of a grand canonical ensemble) is generally different from that in a similar system in which T and N are fixed, i.e. a member of a canonical ensemble. Calculate and interpret the difference.
5.7. For a uniform three-site Ising ring with ferromagnetic coupling (and no external field), in thermal equilibrium at temperature T, calculate the correlation coefficients Ks sksk' for both k = k' and k k' .
5.8.* For a field-free 1D Ising system of N >> 1 “spins”, in thermal equilibrium at temperature T, calculate the correlation coefficient Ks slsl+n, where l and ( l + n) are the numbers of two specific spins in the chain.
Hint: Consider a mixed partial derivative of the statistical sum calculated in Problem 4.21 for an Ising chain with an arbitrary set of Jk, over a part of these parameters.
82 Note that these two cases may be considered as the non-interacting limits of, respectively, the Ising model (4.23) and the classical limit of the Heisenberg model (4.21), whose analysis within the Weiss approximation was the subject of Problems 4.22 and 4.23.
Chapter 5
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5.9. Within the framework of the Weiss molecular-field approximation, calculate the variance of spin fluctuations in the d-dimensional Ising model. Use the result to derive the conditions of quantitative validity of the approximation.
5.10. Calculate the variance of energy fluctuations in a quantum harmonic oscillator with frequency , in thermal equilibrium at temperature T, and express it via the average energy.
5.11. The spontaneous electromagnetic field inside a closed volume V is in thermal equilibrium at temperature T. Assuming that V is sufficiently large, calculate the variance of fluctuations of the total energy of the field, and express the result via its average energy and temperature. How large should the volume V be for your results to be quantitatively valid? Evaluate this limitation for room temperature.
5.12. Express the r.m.s. uncertainty of the occupancy Nk of a certain energy level k by non-interacting:
(i) classical particles,
(ii) fermions, and
(iii) bosons,
in thermodynamic equilibrium, via the level’s average occupancy Nk, and compare the results.
5.13. Write a general expression for the variance of the number of particles in the ideal gases of bosons and fermions, at fixed V, T, and . Spell out the result for the degenerate Fermi gas.
~
5.14. Express the variance of the number of particles, 2
N V, T,, of a single-phase system in
equilibrium, via its isothermal compressibility T –( V/ P) T, N/ V.
5.15.* Calculate the low-frequency spectral density of fluctuations of the pressure P of an ideal classical gas, in thermal equilibrium at temperature T, and estimate their variance. Compare the former result with the solution of Problem 3.2.
Hints: You may consider a cylindrically-shaped container of volume
A
V = LA (see the figure on the right), and start by using the Maxwell N , T
distribution of velocities to calculate the spectral density of the force F ( t) F ( t)
exerted by the confined particles on its plane lid of area A, approximating the force with a delta-correlated process.
L
5.16. Calculate the low-frequency spectral density of fluctuations of the electric q
current I( t) due to the random passage of charged particles between two conducting electrodes – see the figure on the right. Assume that the particles are emitted, at random times, by one of the electrodes, and are fully absorbed by the counterpart electrode. Can your result be mapped onto some aspect of the electromagnetic blackbody radiation?
Hint: For the current I( t), use the same delta-correlated-process approximation as for the force F( t) in the previous problem.
5.17. Perhaps the simplest model of the diffusion is the 1D discrete random walk: each time interval , a particle leaps, with equal probability, to any of two adjacent sites of a 1D lattice with a Chapter 5
Page 42 of 44
SM: Statistical Mechanics
spatial period a. Prove that the particle’s displacement during a time interval t >> obeys Eq. (77), and calculate the corresponding diffusion coefficient D.
5.18.83 A long uniform string, of mass per unit length, is attached to a F t
firm support and stretched with a constant force (“tension”) T – see the figure on T
the right. Calculate the spectral density of the random force F( t) exerted by the
string on the support point, within the plane normal to its length, in thermal equilibrium at temperature T.
Hint: You may assume that the string is so long that transverse waves propagating along it from the support point never come back.
5.19.84 Each of the two 3D isotropic harmonic oscillators, with mass m, resonance frequency 0, and damping coefficient > 0, has the electric dipole moment d = qs, where s is the vector of the oscillator’s displacement from its equilibrium position. Use the Langevin formalism to calculate the average potential of electrostatic interaction (a particular case of the so-called London dispersion force) of these oscillators separated by distance r >> ( T/ m)1/2/0, in thermal equilibrium at temperature T >>
0. Also, explain why the approach used to solve a very similar Problem 2.18 is not directly applicable to this case.
1
2
Hint: You may like to use the following integral:
2
.
2
2
2
0 1
d
a
4 a
5.20.* Within the van der Pol approximation,85 calculate the major statistical properties of small fluctuations of classical self-oscillations (including their linewidth), for: (i) a free (“autonomous”) run of the oscillator, and
(ii) its phase being locked by an external sinusoidal force,
assuming that the fluctuations are caused by a noise with a smooth spectral density Sf().
5.21. Calculate the correlation function of the coordinate of a 1D harmonic oscillator with low damping, in thermal equilibrium. Compare the solution with that of the previous problem.
5.22. A lumped electric circuit consisting of a capacitor C shortened with an Ohmic resistor R is in thermal equilibrium at temperature T. Use two different approaches to calculate the variance of the thermal fluctuations of the capacitor’s electric charge Q. Estimate the effect of quantum fluctuations.
83 This problem, conceptually important for the quantum mechanics of open systems, was also given in Chapter 7
of the QM part of this series.
84 This system, with an arbitrary temperature, was the subject of QM Problem 7.6, with QM Problem 5.15 serving as the background. However, the method used in the model solutions of those problems requires one to prescribe, to the oscillators, different frequencies 1 and 2 at first, and only after this more general problem has been solved, pursue the limit 1 2, while neglecting dissipation altogether. The goal of this problem is to show that the result of that solution is valid even at non-zero damping.
85 See, e.g., CM Secs. 5.2-5.5. Note that in quantum mechanics, a similar approach is called the rotating-wave approximation (RWA) – see, e.g., QM Secs. 6.5, 7.6, 9.2, and 9.4.
Chapter 5
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5.23. Consider a very long, uniform, two-wire transmission line (see the
figure on the right) with a wave impedance Z, which allows the propagation of I
V
TEM electromagnetic waves with negligible attenuation. Calculate the variance I
V2 of spontaneous fluctuations of the voltage V between the wires within a small interval of cyclic frequencies, in thermal equilibrium at temperature T.
Hint: As an E&M reminder,86 in the absence of dispersive materials, TEM waves propagate with a frequency-independent velocity, and with the voltage V and current I (see the figure above) related as V ( x,t)/ I( x,t) = Z, where Z is the line’s wave impedance.
5.24. Now consider a similar long transmission line but terminated, at one end, with an impedance-matching Ohmic resistor R = Z. Calculate the variance V2 of the voltage across the resistor, and discuss the relation between the result and the Nyquist formula (81b), including numerical factors.
Hint: A termination with resistance R = Z absorbs incident TEM waves without reflection.
U ( q)
5.25. An overdamped classical 1D particle escapes from a potential well I w
with a smooth bottom but a sharp top of the barrier – see the figure on the right.
Perform the necessary modification of the Kramers formula (139).
0
q
q
q
1
2
5.26.* Similar particles, whose spontaneous electric dipole moments p have a field-independent magnitude p0, are uniformly distributed in space with a density n so low that their mutual interaction is negligible. Each particle may rotate without substantial inertia but under a kinematic friction torque proportional to its angular velocity. Use the Smoluchowski equation to calculate the complex dielectric constant () of such a medium, in thermal equilibrium at temperature T, for a weak, linearly-polarized rf electric field.
5.27.* Prove that for systems with relatively low inertia (i.e. relatively high damping), at not very high temperatures, the Fokker-Planck equation (149) reduces to the Smoluchowski equation (122) – in the sense described by Eq. (153) and the accompanying statement.
5.28.* Use the 1D version of the Fokker-Planck equation (149) to prove the solution (156) of the Kramers problem.
5.29. A constant external torque, applied to a 1D mechanical pendulum with mass m and length l, has displaced it by angle 0 < /2 from the vertical position. Calculate the average rate of the pendulum’s rotation induced by relatively small thermal fluctuations of temperature T.
5.30. A classical particle may occupy any of N similar sites. Its weak interaction with the environment induces random uncorrelated incoherent jumps from the occupied site to any other site, with the same time-independent rate . Calculate the correlation function and the spectral density of fluctuations of the instant occupancy n( t) (equal to either 1 or 0) of a site.
86 See, e.g., EM Sec. 7.6.
Chapter 5
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Chapter 6. Elements of Kinetics
This chapter gives a brief introduction to the basic notions of physical kinetics. Its main focus is on the Boltzmann transport equation, especially within the simple relaxation-time approximation (RTA), which allows an approximate but reasonable and simple description of transport phenomena (such as the electric current and thermoelectric effects) in gases, including electron gases in metals and semiconductors.
6.1. The Liouville theorem and the Boltzmann equation
Physical kinetics (not to be confused with “kinematics”!) is the branch of statistical physics that deals with systems out of thermodynamic equilibrium. Major effects addressed by kinetics include: (i) for autonomous systems (those out of external fields): the transient processes ( relaxation), that lead from an arbitrary initial state of a system to its thermodynamic equilibrium; (ii) for systems in time-dependent (say, sinusoidal) external fields: the field-induced periodic oscillations of the system’s variables; and
(iii) for systems in time-independent (“dc”) external fields: dc transport.
In the last case, we are dealing with stationary (/ t = 0 everywhere), but non-equilibrium situations, in which the effect of an external field, continuously driving the system out of equilibrium, is partly balanced by its simultaneous relaxation – the trend back to equilibrium. Perhaps the most important effect of this class is the dc current in conductors and semiconductors,1 which alone justifies the inclusion of the basic notions of kinetics into any set of core physics courses.
The reader who has reached this point of the notes already has some taste of physical kinetics because the subject of the last part of Chapter 5 was the kinetics of a “Brownian particle”, i.e. of a
“heavy” system interacting with an environment consisting of many “lighter” components. Indeed, the equations discussed in that part – whether the Smoluchowski equation (5.122) or the Fokker-Planck equation (5.149) – are valid if the environment is in thermodynamic equilibrium, but the system of our interest is not necessarily so. As a result, we could use those equations to discuss such non-equilibrium phenomena as the Kramers problem of the metastable state’s lifetime.
In contrast, this chapter is devoted to the more traditional subject of kinetics: systems of many similar particles – generally, interacting with each other but not too strongly, so the energy of the system still may be partitioned into a sum of single-particle components, with the interparticle interactions considered as a perturbation. Actually, we have already started the job of describing such a system at the beginning of Sec. 5.7. Indeed, in the absence of particle interactions (i.e. when it is unimportant whether the particle of our interest is “light” or “heavy”), the probability current densities in the coordinate and momentum spaces are given, respectively, by Eq. (5.142) and the first form of Eq. (5.143a), so the continuity equation (5.140) takes the form
w
.
(6.1)
q
wq
p
wp 0
t
1 This topic was briefly addressed in EM Chapter 4, avoiding its aspects related to thermal effects.
© K. Likharev
SM: Statistical Mechanics
If similar particles do not interact, this equation for the single-particle probability density w(q, p, t) is valid for each of them, and the result of its solution may be used to calculate any ensemble-average characteristic of the system as a whole.
Let us rewrite Eq. (1) in the Cartesian component form,
w
wq
wp
,
(6.2)
j
j 0
t
j q
p
j
j
where the index j numbers all degrees of freedom of the particle under consideration, and assume that its motion (perhaps in an external, time-dependent field) may be described by a Hamiltonian function H ( qj, pj, t). Plugging into Eq. (2) the Hamiltonian equations of motion:2
H
q
,
p H ,
(6.3)
j
j
p
q
j
j
we get
w
H
w
w H 0.
(6.4)
t
j
q
p
p
q
j
j
j
j
After differentiation of both parentheses by parts, the equal mixed terms w2H/ qj pj and w2H/ pj qj cancel, and using Eq. (3) again, we get the so-called Liouville theorem 3
w
w
w
q
p
0
Liouville
.
(6.5)
j
j
t
theorem
j q
p
j
j
Since the left-hand side of this equation is just the full derivative of the probability density w considered as a function of the generalized coordinates qj( t) of a particle, its generalized momenta components pj( t), and (possibly) time t,4 the Liouville theorem (5) may be represented in a surprisingly simple form:
(
dw q, p, t)
0 .
(6.6)
dt
Physically, this means that the elementary probability dW = wd 3 qd 3 p to find a Hamiltonian particle in a small volume of the coordinate-momentum space [q, p], with its center moving in accordance to the deterministic law (3), does not change with time – see Fig. 1.
q( t), p( t)
Fig. 6.1. The Liouville
t' t
theorem’s
interpretation:
probability’s conservation
t
at the system’s motion flow
d 3 q d 3 p
through the [q, p] space.
2 See, e.g., CM Sec. 10.1.
3 Actually, this is just one of several theorems bearing the name of Joseph Liouville (1809-1882).
4 See, e.g., MA Eq. (4.2).
Chapter 6
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SM: Statistical Mechanics
At first glance, this fact may not look surprising because according to the fundamental Einstein relation (5.78), one needs non-Hamiltonian forces (such as the kinematic friction) to have diffusion. On the other hand, it is striking that the Liouville theorem is valid even for (Hamiltonian) systems with deterministic chaos,5 in which the deterministic trajectories corresponding to slightly different initial conditions become increasingly mixed with time.
For an ideal gas of 3D particles, we may use the ordinary Cartesian coordinates rj (with j = 1, 2, 3) as the generalized coordinates qj, so pj become the Cartesian components mvj of the usual (linear) momentum, and the elementary volume is just d 3 rd 3 p – see Fig. 1. In this case, Eqs. (3) are just p
j
r
v , p ,
(6.7)
j
j
j
F j
m
where F is the force exerted on the particle, so the Liouville theorem may be rewritten as 3
w
w
w
v
F
0
,
(6.8)
j
j
t
j
r
1
p
j
j
and conveniently represented in the vector form
w
v w F w 0 .
(6.9)
t
r
p
Of course, the situation becomes much more complex if the particles interact. Generally, a system of N similar particles in 3D space has to be described by the probability density being a function of (6 N + 1) arguments: 3 N Cartesian coordinates, plus 3 N momentum components, plus time. An analytical or numerical solution of any equation describing the time evolution of such a function for a typical system of N ~ 1023 particles is evidently a hopeless task. Hence, any theory of realistic systems’
kinetics has to rely on making reasonable approximations that would simplify the situation.
One of the most useful approximations (sometimes called Stosszahlansatz – German for the
“collision-number assumption”) was suggested by Ludwig Boltzmann for a gas of particles that move freely most of the time but interact during short time intervals, when a particle comes close to either an immobile scattering center (say, an impurity in a conductor’s crystal lattice) or to another particle of the gas. Such brief scattering events may change the particle’s momentum. Boltzmann argued that they may be still approximately described Eq. (9), with the addition of a special term (called the scattering integral) to its right-hand side:
Boltzmann
w
w
transport
v w F w
.
(6.10)
r
p
scattering
equation
t
t
This is the Boltzmann transport equation, sometimes called just the “Boltzmann equation” for short. As will be discussed below, it may give a very reasonable description of not only classical but also quantum particles, though it evidently neglects the quantum-mechanical coherence/entanglement effects6 –
besides those that may be hidden inside the scattering integral.
5 See, e.g., CM Sec. 9.3.
6 Indeed, the quantum state coherence is described by off-diagonal elements of the density matrix, while the classical probability w represents only the diagonal elements of that matrix. However, at least for the ensembles close to thermal equilibrium, this is a reasonable approximation – see the discussion in Sec. 2.1.
Chapter 6
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Essential Graduate Physics
SM: Statistical Mechanics
The concrete form of the scattering integral depends on the type of particle scattering. If the scattering centers do not belong to the ensemble under consideration (an example is given, again, by impurity atoms in a conductor), then the scattering integral may be expressed as an evident generalization of the master equation (4.100):
w
3
d p'
w r p ' t
w r p t ,
(6.11)
g
scatteerin
( , , )
( , , )
p ' p
p p
'
t
where the physical sense of pp ’ is the rate (i.e. the probability per unit time) for the particle to be scattered from the state with the momentum p into the state with the momentum p ’ – see Fig. 2.
scattering
center
p '
Fig. 6.2. A single-particle scattering event.
p
Most elastic interactions are reciprocal, i.e. obey the following relation (closely related to the reversibility of time in Hamiltonian systems): pp ’ = p ’p, so Eq. (11) may be rewritten as7
w
3
d p'
w r p ' t w r
t
g
scatteerin
( , , ) ( , , )
.
(6.12)
t
p p
'
p
With such scattering integral, Eq. (10) stays linear in w but becomes an integro-differential equation, typically harder to solve analytically than differential equations.
The equation becomes even more complex if the scattering is due to the mutual interaction of the particle members of the system – see Fig. 3.
p ' '
interaction
p '
region
p'
Fig. 6.3. A particle-particle scattering event.
p
In this case, the probability of a scattering event scales as a product of two single-particle probabilities, and the simplest reasonable form of the scattering integral is8
7 One may wonder whether this approximation may work for Fermi particles, such as electrons, for whom the Pauli principle forbids scattering into the already occupied state, so for the scattering p p ’, the term w(r, p, t) in Eq. (12) has to be multiplied by the probability [1 – w(r, p ’, t)] that the final state is available. This is a valid argument, but one should notice that if this modification has been done with both terms of Eq. (12), it becomes w
3
d p'
w r p ' t 1 w r p t w r p t 1 w r ' t g
scatteerin
( , , )
( , , )
( , ,
)
( , , )
.
t
p p
'
p
Opening both square brackets, we see that the probability density products cancel, bringing us back to Eq. (12).
8 This was the approximation used by L. Boltzmann to prove the famous H-theorem, stating that the entropy of the gas described by Eq. (13) may only grow (or stay constant) in time, dS/ dt 0. Since the model is very approximate, that result does not seem too fundamental nowadays, despite all its historic significance.
Chapter 6
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SM: Statistical Mechanics
(
w r, p ' , t) (
w r, p ' , t)
p
p
'
' , '
w
3
3
'
p
'
p
d p' d p
.
(6.13)
g
scatteerin
'
t
(
w r, p, t) (
w r, p , t)
pp ' ,
'
'
'
p
'
p
The integration dimensionality in Eq. (13) takes into account the fact that due to the conservation of the total momentum at scattering,
p p p ' p ' ,
(6.14)
'
'
one of the momenta is not an independent argument, so the integration in Eq. (13) may be restricted to a 6D p-space rather than the 9D one. For the reciprocal interaction, Eq. (13) may also be a bit simplified, but it still keeps Eq. (10) a nonlinear integro-differential transport equation, excluding such powerful solution methods as the Fourier expansion – which hinges on the linear superposition principle.
This is why most useful results based on the Boltzmann transport equation depend on its further simplifications, most notably the relaxation-time approximation – RTA for short.9 This approximation is based on the fact that in the absence of spatial gradients ( = 0), and external forces (F = 0), in the thermal equilibrium, Eq. (10) yields
w
w
,
(6.15)
scattering
t
t
so the equilibrium probability distribution w 0(r, p, t) has to turn any scattering integral to zero. Hence at a small deviation from the equilibrium,
~
w(r, p, t) (
w r, p, t) w (r, p, t) 0 , (6.16)
0
the scattering integral should be proportional to the deviation w
~ , and its simplest reasonable model is
Relaxation-
time
w
w
~
,
(6.17)
approximation
t
g
scatteerin
(RTA)
where is a phenomenological constant (which, according to Eq. (15), has to be positive for the system’s stability) called the relaxation time. Its physical meaning will be more clear in the next section.
The relaxation-time approximation is quite reasonable if the angular distribution of the scattering rate is dominated by small angles between vectors p and p ’ – as it is, for example, for the Rutherford scattering by a Coulomb center.10 Indeed, in this case the two values of the function w participating in Eq. (12) are close to each other for most scattering events, so the loss of the second momentum argument (p ’) is not too essential. However, using the Boltzmann-RTA equation that results from combining Eqs. (10) and (17),
w
w
~
Boltzmann-
v w F w ,
(6.18)
RTA
t
r
p
equation
we should always remember that this is just a phenomenological model, sometimes giving completely wrong results. For example, it prescribes the same time scale () to the relaxation of the net momentum 9 Sometimes this approximation is called the “BGK model”, after P. Bhatnager, E. Gross, and M. Krook who suggested it in 1954. (The same year, a similar model was considered by P. Welander.) 10 See, e.g., CM Sec. 3.7.
Chapter 6
Page 5 of 38
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of the system, and to its energy relaxation, while in many real systems, the latter process (that results from inelastic collisions) may be substantially longer. Naturally, in the following sections, I will describe only those applications of the Boltzmann-RTA equation that give a reasonable description of physical reality.
6.2. The Ohm law and the Drude formula
Despite its shortcomings, Eq. (18) is adequate for quite a few applications. Perhaps the most important of them is deriving the Ohm law for dc current in a “nearly-ideal” gas of charged particles, whose only important deviation from ideality is the rare scattering effects described by Eq. (17). As a result, in equilibrium it is described by the stationary probability w 0 of an ideal gas (see Sec. 3.1): g
w (r, p, t)
N
(6.19)
0
,
2
3
where g is the internal degeneracy factor (say, g = 2 for electrons due to their spin), and N() is the average occupancy of a quantum state with momentum p, that obeys either the Fermi-Dirac or the Bose-Einstein distribution:
1
N
.
(6.20)
exp
,
/ T
p
1
(The following calculations will be valid, up to a point, for both statistics and hence, in the limit / T
–, for a classical gas as well.)
Now let a uniform dc electric field E be applied to a uniform gas of similar particles with electric charge q, exerting the force F = q E on each of them. Then the stationary solution of Eq. (18), with / t = 0, should also be stationary and spatially uniform ( r = 0), so this equation is reduced to w
~
q E w .
(6.21)
p
Let us require the electric field to be relatively low, so that the perturbation w
~ it produces is relatively
small, as required by our basic assumption (16).11 Then on the left-hand side of Eq. (21), we can neglect that perturbation, by replacing w with w 0, because that side already has a small factor (E). As a result, this equation yields
w
~
w
q E w
q E
,
(6.22)
p
p 0
0
where the second step implies isotropy of the parameters and T, i.e. their independence of the direction of the particle’s momentum p. But the gradient p is nothing else than the particle’s velocity 11 Since the scale of the fastest change of w 0 in the momentum space is of the order of w 0/ p = ( w 0/)( d/ dp) ~
(1/ T) v, where v is the particle speed scale, the necessary condition of the linear approximation (22) is e E << T/ v, i.e. if e E l << T, where l v has the meaning of the effective mean free path. Since the left-hand side of the last inequality is just the average energy given to the particle by the electric field between two scattering events, the condition may be interpreted as the smallness of the gas’ “overheating” by the applied field. However, another condition is also necessary – see the last paragraph of this section.
Chapter 6
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SM: Statistical Mechanics
v – for a quantum particle, its group velocity.12 (This fact is easy to verify for the isotropic and parabolic dispersion law, pertinent to classical particles moving in free space,
p 2
p 2 p 2 p 2
p
1
2
3
.
(6.23)
2 m
2 m
Indeed, in this case, the j th Cartesian components of the vector p is
p
v ,
(6.24)
p
j
j
j
p
m
j
so p = v.) Hence, Eq. (22) may be rewritten as
w
~
w
q E
0
v
.
(6.25)
Let us use this result to calculate the electric current density j. The contribution of each particle to the current density is qv, so the total density is
j q wd 3
v
p q
w w~
v
.
(6.26)
0
d 3 p
Since in the equilibrium state (with w = w 0), the current has to be zero, the integral of the first term in the parentheses has to vanish. For the integral of the second term, plugging in Eq. (25), and then using Eq. (19), we get
2
Sommerfeld
w
gq
N
theory’s
2
j q vE v
0
3
d p
v E v
2
d p dp
(6.27)
result
,
2
3
where d 2 p is the elementary area of the constant energy surface in the momentum space, while dpis the momentum differential’s component normal to that surface. The real power of this result13 is that it is valid even for particles with an arbitrary dispersion law (p) (which may be rather complicated, for example, for particles moving in space-periodic potentials14), and gives, in particular, a fair description of conductivity’s anisotropy in crystals.
For free particles whose dispersion law is isotropic and parabolic, as in Eq. (23), the constant energy surface is a sphere of radius p, so d 2 p = p 2 d = p 2sin d d, while dp = dp. In the spherical coordinates, with the polar axis directed along the electric field vector E, we get (Ev) = E v cos. Now separating the vector v outside the parentheses into the component v cos directed along the vector E, and two perpendicular components, v sincos and v sinsin, we see that the integrals of the last two components over the angle give zero. Hence, as we could expect, in the isotropic case the net current is directed along the electric field and obeys the linear Ohm law, Ohm
j
law
E ,
(6.28)
12 See, e.g., QM Sec. 2.1.
13 It was obtained by Arnold Sommerfeld in 1927.
14 See, e.g., QM Secs. 2.7, 2.8, and 3.4. (In this case, p should be understood as the quasimomentum rather than the genuine momentum.)
Chapter 6
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SM: Statistical Mechanics
with a field-independent, scalar15 electric conductivity
2
2
gq
N
2
2
2
d
sin d cos θ p dp v
.
(6.29)
2 3
0
0
0
(Note that is proportional to q 2 and hence does not depend on the particle charge sign.16) Since sin d is just – d(cos), the integral over equals (2/3). The integral over d is of course just 2, while that over p may be readily transformed to one over the particle’s energy (p) = p 2/2 m: p 2 =
2 m, v 2 = 2/ m, p = (2 m)1/2, so dp = ( m/2)1/2 d, and p 2 dpv 2 = (2 m)( m/2)1/2 d (2/ m) (8 m3)1/2 d. As a result, the conductivity equals
2
gq 4
N
1/ 2
8
m
d .
(6.30)
3
3
2
3
0
Now we may work out the integral in Eq. (30) by parts, first rewriting [- N()/] d as – d[ N()]. Due to the fast (exponential) decay of the factor N() at , its product by the factor (8 m3)1/2 vanishes at both integration limits, and we get
2
gq 4
N d
gq
m
m
N
d
3
2
8
3 1/2
4 8 1/2
3 1/2
2 3
0
2 3 3
2
0
(6.31)
2
3 / 2
q
gm
N 1/ 2
d .
m
2 2 3
0
But according to Eq. (3.40), the last factor in this expression (following the sign) is just the particle density n N/ V, so Sommerfeld’s result is reduced, for an arbitrary temperature and any particle statistics, to the very simple Drude formula,17
q
2
n ,
(6.32) Drude
m
formula
which should be well familiar to the reader from an undergraduate physics course.
As a reminder, here is its simple classical derivation.18 Let be the average time after the last scattering event that has caused particles to lose the deterministic component of their velocity, vdrift, provided by the electric field E on the top of the particle’s random thermal motion – which does not contribute to the net current. Using the 2nd Newton law to describe the particle’s acceleration by the 15 As Eq. (27) shows, if the dispersion law (p) is anisotropic, the current density direction may be different from that of the electric field. In this case, conductivity should be described by a tensor jj’, rather than a scalar.
However, in most important conducting materials, the anisotropy is rather small – see, e.g., EM Table 4.1.
16 This is why the Hall effect, which lacks such ambivalence (see, e.g., QM 3.2), is frequently used to determine the dominating type of charge carriers in semiconductors: electrons or holes, see Sec. 4 below.
17 It was derived in 1900 by Paul Drude. Note that Drude also used the same arguments to derive a very simple (and very reasonable) approximation for the complex electric conductivity in the ac field of frequency : () =
(0)/(1 – i), with (0) given by Eq. (32); sometimes the name “Drude formula” is used for this expression. Let me leave its derivation, from the Boltzmann-RTA equation, for the reader’s exercise.
18 See also EM Sec. 4.2. Note that the frequently met definition of as the “the average time interval between two sequential scattering events” would lead to an extra factor of ½ in the expressions for vdrift and .
Chapter 6
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SM: Statistical Mechanics
field, dv/ dt = q E/ m, we get vdrift = q E/ m. Multiplying this result by the particle’s charge q and density n N/ V, we get the Ohm law j = E, with given by Eq. (32).
Sommerfeld’s derivation of the Drude formula poses an important conceptual question. The structure of Eq. (30) implies that the only quantum states contributing to the electric conductivity are those whose derivative [- N()/] is significant. For the Fermi particles such as electrons, in the limit T << F, these are the states at the very Fermi surface. On the other hand, Eq. (32) and the whole Drude reasoning, involve the density n of all electrons. So, what exactly electrons are responsible for the conductivity: all of them, or only those at the Fermi surface? For the resolution of this paradox, let us return to Eq. (22) and analyze the physical meaning of that result. Let us compare it with the following model distribution:
~
w
w (r, p p, t) ,
(6.33)
model
0
where p~ is some time-independent, small vector that describes a small shift of the unperturbed distribution w 0 as a whole, in the momentum space. Performing the Taylor expansion of Eq. (33) in this small parameter, and keeping only two leading terms, we get
~
~
~
w
w (r, p, t) w
,
with w
p w (r, p, t) .
(6.34)
model
0
model
model
p
0
Comparing the last expression with the first form of Eq. (22), we see that they coincide if p~ E
q τ F .
(6.35)
This means that Eq. (22) describes a small shift of the equilibrium distribution of all particles (in the momentum space) by q E along the electric field’s direction, justifying the cartoon shown in Fig. 4.
p
(a)
p
(b)
2
2
F q E
Fig. 6.4. Filling of momentum states by
0
p
p
a degenerate electron gas: (a) in the
1
1
absence and (b) in the presence of an
external electric field E. Arrows show
representative scattering events.
p~ F
At E = 0, the system is in equilibrium, so the quantum states inside the Fermi sphere ( p < p F), are occupied, while those outside of it are empty – see Fig. 4a. Electron scattering events may happen only between states within a very thin layer ( p 2/2 m – F ~ T) at the Fermi surface because only in this layer the states are partially occupied, so both components of the product w(r, p, t)[1 – w(r, p ’, t)], mentioned in Sec. 1, do not vanish. These scattering events, on average, do not change the equilibrium probability distribution, because they are uniformly spread over the Fermi surface.
Now let the electric field be turned on instantly. Immediately it starts accelerating all electrons in its direction, i.e. the whole Fermi sphere starts moving in the momentum space, along the field’s direction in the real space. For elastic scattering events (with p ’ = p ), this creates an addition of occupied states at the leading edge of the accelerating sphere and an addition of free states on its trailing Chapter 6
Page 9 of 38