by
Benjamin Ira Ross A. B., Harvard University
(1971)
SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY February, 1976
Signature of Author:
Signature redacted
Department of Physics, October 6, 1975
Signature redacted
Certified by:
Accepted by:
Thesis Supervisor
Signature redacted
Chairman, Departmental Comrnittee on Graduate Students
ARCHIVES
VAASS IT 1
t 9
4' 1
by
Benjamin Ira Ross
Submitted to the Department of Physics on October 6, 1975, in partial fulfillment of the requirements for the degree of Doctor of Philosophy.
ABSTRACT
The ballast resistor is an electrical device which was widely used as a constant current source prior to the development of electronic
amplifiers. The basis of its operation is a thermal instability which appears far from equilibrium.
We study this instability and the ensuing spatial structure as an example of non-equilibrium thermodynamics. The steady-state solutions of the energy conservation equation show a first order phase transition.
The steady states of this system correspond to the minima of a functional we denote l. Away from the steady state, X always
decreases over time. Absolute minima of I correspond to stable states and relative minima correspond to metastable states.
When the spectrum of fluctuations about the uniform steady state is calculated from the fluctuation-dissipation relation, it is found that the probability of a given state is given by e-l/c, where c is a constant
depending only on values measured at the steady state. We conjecture that this formula applies everywhere.
Since the ballast resistor is a one-dimensional system, this conjecture implies that it will lack long-range order. Another independent calculation verifies this consequence.
We also calculate that the frequency response of a ballast resistor operating in the two-phase region is a Lorentzian.
Experimental results showing the coexistence curve of a ballast resistor are presented.
Thesis Supervisor: J. D. Litster Title: Professor of Physics
ACKNOWLEDGEMENTS
I would like to express my deep appreciation to Professor Pierre Aigrain, who suggested the topic of this research and whose
advice and encouragement were invaluable. I would equally like to thank my advisor, Professor J. D. Litster, for his advice and guidance.
I am grateful to Professor B. Patton, Professor M. Stephen, Mr. C. Guttel, and Dr. K. Kitahara for helpful discussions. I would also like to thank Professor George Benedek for his assistance and encouragement in undertaking this study, and for his invaluable
instruction in the importance of getting to the essentials of a problem. I further thank Professor J. Bok and Dr. J. P. Maneval for their hospitality at the Ecole Normale Superieure, Paris, where a part of this work was done.
TABLE OF CONTENTS
Page
I. INTRODUCTION . . . . . . .
References . . . .
II. MACROSCOPIC DESCRIPTION OF THE STEADY STATE
5 10 11 A. Stability . . . . . B. Critical Behavior . . . . C. Potential Function . . . . References . . . . III. FLUCTUATIONS . . . .
A. Potentials and Distribution Functions . . B. Fluctuations Near the Uniform Steady State C. Critical Fluctuations . . . . D. Potential of Fluctuations . . . .
References . . . .
IV. LONG-RANGE ORDER . . . .
References . . . .
V. DYNAMIC RESPONSE IN THE TWO-PHASE REGION A. Response Time . . . .
B. Critical Slowing Down . . . .
VI. EXPERIMENTS . . . . References . . . . VII. CONCLUSION . . . . 0 . 11 . . 15 . . 16 . . 21 . . 22 . . 22 . 23 . . 32 . . 34 . . 38 . . 39 . . 47 . . 48 . . 48 . . 56 60 72 73
Note: Chapters V and VII have no references.
.
The field of irreversible thermodynamics arose in the years before and after World War II as a study of systems which were close enough to equilibrium that their behavior could be described by linear relationships between fluxes and driving forces. With the Onsager reciprocal relations and Prigogine's rule of minimum entropy
production, 2 the fundamental laws governing the macroscopic behavior of these linear non-equilibrium systems seemed to be understood.
Attention gradually turned toward systems operating outside the linear range.
It became apparent in the sixties that nonlinear effects could not always be described by adding additional power series terms to the linear equations. In many systems, as one moves away from equilibrium a symmetry-breaking instability occurs at some well-defined point. Beyond this point, the system usually exhibits a spatial or temporal
structure (called a "dissipative structure"3) which is maintained by the dissipation of energy and decays away when the dissipation ceases. These instabilities have been the subject of considerable study in recent years. 3, 4
Once these concepts had been elaborated, it became clear that they covered a number of instabilities which had previously been studied by physicists. Most notable were the instabilities in fluid flows which lead to more complex flow patterns or to turbulence. Closely related is the formation of convection cells in a layer of fluid heated from below
(the Benard problem). 5 These had been the subject of research for many decades. The onset of laser action in a cavity was another such
instabilities6,7,8 have been studied; the analogy between one electrical system and a magnetic phase transition was pointed out as long ago as 1905. 9
The most complex and most interesting dissipative structures are living things. Their decay toward equilibrium when the source of
energy is cut off is well known to the casual observer and has been thoroughly explored without the aid of irreversible thermodynamics. But the manner in which life arose, and the thermodynamic restraints on ongoing metabolic processes, are very poorly known at present.
Perhaps the most ambitious aim that may be set for this field of study is the understanding of living things and of the world eco-system as
dissipative processes. In other words, to explain the thermodynamic basis of the origin of life, of Darwinian evolution, and of plant and animal metabolism. The purpose of this study is to contribute in a small way to the advance toward this goal by extending the knowledge of simple dissipative structures.
In equilibrium thermodynamics, the free energy function provides us with a general theory with which we can characterize the mean state of the system and also find the complete distribution function giving the probability of all possible states of the system. The distribution function, given by e-G , is very sharply peaked (except near a thermodynamic
critical point) and the most likely state, found by minimizing the free energy G, is then identical with the mean state. Stable and metastable states correspond respectively to absolute and relative minima of G.
At present, there is no analogous general theory describing instabilities in dissipative systems. We may define a steady state of a system with constant boundary conditions as a state whose state variables,
time. (This definition permits regular oscillations, and in some cases may permit turbulence. ) The steady state is found by solving the
macroscopic rate equations and conservation equations describing a given system with all time derivatives set equal to zero (except those differentiating regular oscillations, if any). For example, for the fluid heated from below, the rate equations relate conductive heat flow to thermal gradients and fluid velocity fields to the buoyant force due to thermal expansion of the water. The conservation laws in this case are those of matter, energy, and momentum. The rate equations, if they are linear, will involve dissipative rate constants which may depend on other system variables. In this example, the thermal conductivity and viscosity may depend on temperature.
Solutions of this system of equations may be stable, unstable, or metastable. Unstable solutions may be distinguished by the well-known method of small perturbations, or by the closely related method of Lyapunov functions. Glansdorff and Prigogine3 have produced a Lyapunov function of very wide applicability. The identification of metastable and absolutely stable states is more difficult. The simplest method in concept is to find the probability distribution function of all states; the stable state will be more probable than the metastable state. This is, however, very difficult in practice and has been possible only in extremely simple cases.
Another method is to treat the growth of perturbations which are large in amplitude but of small extent in wave vector space or in real space. This method has been used in fluid mechanics, where stable states which become metastable with increasing Reynolds number are
said to have "soft instabilities".10 In fluid mechanics, periodic pertur-bations of a given wave vector are used; we shall use spatially localized perturbations in our analysis.
Of course the complete description of the steady-state behavior of a system is provided by the distribution function giving the probability that the system is in any particular state at any moment. Such functions have as yet been obtained only in a very small number of cases, which may be classified according to the dimensionality of the space occupied by the state variables of the system. The entire distribution function has been obtained only in such zero-dimensional cases as the tunnel diode7 and the single-mode laser 1 in which the state of the system is described by a single scalar variable (current and field amplitude
respectively). In the one-dimensional case of the laser with continuously many modes4 and the three-dimensional case of the fluid heated from below, 1 distribution functions have been obtained only in the immediate vicinity of the critical point. These two systems possess critical points which lie on an axis of symmetry of their phase diagrams, which
resemble ferromagnetic phase diagrams in this respect. The critical points of these systems are the points at which lasing or convection
begins, when there is no imposed electromagnetic or velocity field. We shall be studying a one-dimensional system which lacks this symmetry, making its phase diagram more like that of a liquid-vapor system. For this system we shall present a distribution function over all states. Its validity will be proven only in the vicinity of the uniform steady state; however, the conjecture that it is valid for all states and for any given boundary conditions will be strongly motivated.
Up to now two approaches have been used to derive distribution functions. In the master equation method, one starts with equations expressing the rates of transition between the various microstates of the system. The second approach is to add Langevin-type fluctuating terms to the macroscopic equations describing the problem. Well-known
mathematical methods are often used to transform these into Focker-Planck type equations. We shall employ the latter method, and
supplement it with an exploration of the relationship between the distribu-tion funcdistribu-tion and quantities derivable from the usual macroscopic
REFERENCES (Chapter I)
1. L. Onsager, Phys. Rev. 37, 45 (1931); 38, 2265 (1931).
2. I. Prigogine, Introduction to Thermodynamics of Irreversible Processes (Interscience, New York, 1967).
3. P. Glansdorff and I. Prigogine, Thermodynamic Theory of Structure, Stability, and Fluctuations (Wiley-Interscience, London, 1971). 4. H. Haken, Rev. Mod. Phys. 47, 67 (1975).
5. For a recent compilation of literature see M. G. Velarde, in
R. Balian, ed., Hydrodynamics (Gordon and Breach, New York, 1975).
6. H. Busch, Stabilitlt, Labilitst und Pendulungen in der Elektro-technik (S. Hirzel, Leipzig, 1913).
7. R. Landauer, J. Appl. Phys. 33, 2209 (1962).
8. P. M. Horn, T. Carruthers, and M. T. Long, to be published. 9. H. Th. Simon, Phys. Z. 6, 297 (1905).
10. A. S. Monin and A. M. Yaglom, Statistical Fluid Mechanics (M. I. T., Cambridge, 1971).
11. H. Risken, Z. Phys. 186, 85 (1965).
II. MACROSCOPIC DESCRIPTION OF THE STEADY STATE
A. Stability
In the late 1890's, Walther Nernst introduced a new kind of incan-descent lamp with a spectrum considerably closer to daylight's than that of the then-current carbon filament. The incandescent element was a ceramic "glower" composed of rare-earth oxide.
The glower's resistance declined rapidly with increasing tempera-ture. If supplied by a constant voltage source, the current would run
away and burn out the glower whenever the supply voltage exceeded a value which was quite close to the operating point. In order to prevent burnouts and to stabilize the operating point against voltage fluctuations, the lamp had to be put in series with a resistance whose load line
crossed the lamp's characteristic at a considerable angle. A sufficiently large constant resistance would consume several times more power than the lamp itself. In order to save power, a material with a very high temperature coefficient of resistance was employed. One thus obtained
dV
a large value of
-
near the operating point with a much lower absolute resistance V/i, thereby greatly reducing the power consumption.The device actually chosen was a thin iron wire in a tube of
hydrogen. It appears that the Nernst lamp was the first application of this procedure for current stabilization. 2
It was only two decades later that Busch3 pointed out that the iron
ballast resistance itself displayed an instability. (The temperature inhomogeneities explained by Busch had previously been noticed, but had been ascribed to variations in the diameter of the wire. 2)
The instability arises as follows. Consider a straight horizontal wire in a tube of gas. The outside of the tube is held at a constant
temperature. When a voltage is applied to the wire, its temperature increases. The wire will lose heat to the outside at a rate A (per unit length). Given the description of the tube of gas, A is a function only of the local temperature T of the wire. The steady state of a wire
constrained to be at uniform temperature is then described by:
V =iR A= i2R
where R is the resistance of the wire per unit length, i is the current, and V is the voltage drop across the wire divided by its length. This can be expressed as:
i = V = (AR)1 (2.1)
Both A and R, and therefore also V, are monotonically increasing functions of temperature. However, if in any temperature range R increases proportionally faster than A, i will be a decreasing function of
T, and the corresponding portion of the i-V characteristic will have a negative slope. Ferromagnetic materials have large dR/dT near their
Curie points; however, the model describes a superconducting wire near its transition temperature equally well. 4
Now the "negative resistance" part of the characteristic, like the analogous negative compressibility region of the equation of state of a van der Waals fluid, is unstable. The wire thus divides into two phases of different temperature. As the voltage is increased, the current
remains constant while the length of the high-temperature region increases.
Our analog of the Maxwell construction for locating the phase transition in the van der Waals fluid can be derived from the energy
that in each region of uniform temperature a and b the rate of heat conduction along the wire W must vanish. We can therefore write:
W2(a) - W2 (b) = 0 or:
b
WdW = 0
We substitute W = -XdT/dx, where X(T) is the thermal conductivity of the wire multiplied by its cross section:
dW
We shall assume here and throughout this study that the heat flow A out of the wire is a function only of the local temperature T(x) of the wire, and does not depend on dT/dx or on the temperature at neighboring points. It is well known that the same is true of the electrical resistance R(T). The local equation for energy conservation may then be written:
d = i2R(T) - A(T) (2.2)
We substitute this into the previous equation to obtain our final result: T
T (A - i dT =R) 0 (2.3)
T a
Since a and b must correspond to points of the characteristic with the same current, there is only one such pair of points which will satisfy this condition, and i will indeed be constant in the two-phase region. We will henceforth define Ta, T and i* to be the temperatures and current which satisfy (2. 3).
If one plots i2 against T, the curve i2 = A(T)/R(T) corresponds to the uniform steady states of the system, including the metastable and
.2 .*2
unstable ones. The line i = i will cut off equal areas above and below, like Maxwell's construction for the van der Waals gas, if one uses a
weighting function XR. This relation was found by Busch3 for the case where X does not depend on temperature.
It is of some theoretical interest to note that the line of unstable steady states of the system is unambiguously defined in terms of the experimentally measurable functions A and R. There can thus be no question of its existence.
In fluid mechanics, as was mentioned in the introduction, the
method traditionally used to distinguish stable from metastable states has been the study of the growth or decay of large-amplitude fluctuations which occupy a small region in real or reciprocal space. The Maxwell
construction approach used above seems at first to be quite different. However, the two lines of argument are easily shown to be equivalent, as follows.
For non-steady state conditions, the energy conservation equation becomes:
- 2 R - A - c bI
(2.4)
where cv is the heat capacity per unit length of the wire. Now suppose we have a ballast resistor which is operating in the metastable region with a current i somewhat higher than i*. The temperature of the wire is T1 > T . We now assume that a nucleus of phase b has formed
some-a a
where along the wire. This is a region of temperature Tb (i(Tb)= i(T1)) much longer than an interface between the two phases, but much shorter than the wire. The length of this nucleus will change at a speed 2u;
positive u means the nucleus is growing. We assume that the interface between the two phases does not change shape as it moves. The rate of temperature change at any point on either side of the nucleus will then be given by:
OT IfT
=u
giving us:
=W 2 R -A -u cv 10
If this version of the conservation equation is used in the derivation of the Maxwell construction, Eq. (2. 3) becomes:
S(A - i2R) dT u b X cv dT (2.5)
a a
Since i is greater than i*, the integral on the left-hand side is negative. As the right-hand integral is positive, u will be positive and the nucleus
will grow. For i < i*, the left-hand integral is negative and the nucleus contracts.
B. Critical Behavior
It is of interest to point out that this system will exhibit a critical point. For real wires, dR/dT is a function with some maximum value at the Curie point. This value is entirely independent of the temperature T of the outside of the tube. If T is increased toward the wire temperature
T, however, we eventually reach a region where heat is transported
purely by conduction and A = k(T3/ 2 _ T3/2). 0 This relation follows from the T1/ 2 dependence of thermal conductivity in an ideal gas. (Busch3
differences there is a linear relationship between A and T - T , and then:
I 6A cc (T - T )00*
A 1T
-For T sufficiently close to T, this quantity can always be made to exceed the maximum value of R~ 1 dR/dT. For some sufficiently small value of T - T one reaches a critical point at which the phase separation just
0
fails to occur. At the critical value of T , the functions A, X, and R should normally be analytic and therefore the critical exponents will have the mean field values given by the Landau theory.
In this argument, we have of course ignored fluctuations, as is done in the equilibrium mean field theory. When the fluctuations are estimated numerically below (Section III. C), they will be seen to be negligible in experimentally accessible cases.
An ideal ferromagnetic wire will have a logarithmic or stronger singularity in dR/dT at the Curie point. 5 The critical point of the ballast resistor will occur in the region near the Curie point where dR/dT is a maximum. As a result of the divergence of dR/dT and the fact it is not an analytic function at the Curie point, we might expect different critical behavior of the ballast resistor from that predicted by the Landau theory. If dR/dT indeed becomes infinite at Tc, the critical point of the ballast resistor will occur at T = T with i = 0.
C. Potential Function
To derive a potential function which is minimized by any steady state of the system, we start from the energy conservation equation (2. 4), written for the steady state:
A(T) - i2R(T) = x (2.6)
This equation has the same form as Newton's second law, with the analogies:
F A - i2R m 4 1
dx 4 XdT dt - dx
We can therefore immediately write the analog of Hamilton's principle: any solution of (2. 6) must minimize the quantity
Tx 2 dT~
2j dx X(T) [A(T')-i2R(T')]dT'+} [(x) x) -wire
(2.7)
The analogy to Hamilton's principle guarantees by itself that any local minimum of this function is a solution of (2. 6). Additionally, since the first term in the integral has the same form as the left-hand side of Eq. (2. 3), we can conclude that absolute minima of (2. 7) correspond to stable solutions of Eq. (2. 4) and relative minima correspond to meta-stable solutions. As can be seen from the derivation of Eq. (2. 3), the two terms in the potential become equal at the steady state if we choose
the lower limit of the temperature integral to be Ta, like the two terms of the van der Waals - Cahn and Hilliard interfacial free energy 7 at equilibrium.
dx L dT dX dT (dT 2 2 dTd (dT
dt d~x [A -i 2R Id-+ >1 TT- \d dxl + id X\T Tdx
)0
We substitute from Eq. (2. 4):
dZ L
rdTK
dT +d (dT\ dx dt-
cv
dtdX+CT
+ + d dT 7 L- dX dT dT d2T]] -jTT -dx + dtdxJ L f d 2 +~Td( dT)+ Xd~d d dTy~ 0 dx-
d dtd d = -ddx + JRXdT + x d dxdT -cv(T)2
dx + X2 T T]Since dT/dt = 0 at the boundaries, the second term vanishes and we have:
: 0 always (2.8)
with the equality holding only when dT/dt = 0 everywhere; i. e., in the steady state.
, is thus a function which is minimum at the steady state and always declines over time for non-steady states. Such functions are known as Lyapunov functions, and their existence is used to demonstrate the stability of solutions of differential equations. Lyapunov functions have been extensively applied to non-equilibrium thermodynamics by
Glansdorff and Prigogine. 8 However, the functions they used, unlike our Z:, are Lyapunov functions only in the vicinity of the steady state. Far away from the steady state, they can either increase or decrease over time.
Glansdorff and Prigogine place considerable emphasis on
deriving Lyapunov functions with the dimensions of entropy production. In view of the interest that has been aroused in entropy production, we will connect it with our potential function, 1>. If we multiply Eq. (2. 6) by T2 and make the analogies:
dx 4 2 dT
dt 4 X~ T-2 dx
our analog of Hamilton' s principle will minimize:
T -1 2
dx XA(T') T, 2 [A(TI) - i 2R(Tv)] d(T1~ + 'FT 2 d (T ) 1
~ T2~T L dx J J
(2. 9) The problem with this expression is that the variation must be carried out at constant CL -1 T-2 dx. For a wire of constant length, this will
introduce additional terms in the variation which will be proportional to the potential at the endpoints. Now if we are interested only in the
transition between two infinitely long single-phase regions, this difficulty can be avoided. If the lower limit of the temperature integral is Ta, the potential vanishes at endpoints located within the single phase regions.
The additional terms in the variation thus vanish. In this case the potential (2. 9) at the steady state is equal to the entropy production due
to heat conduction within the wire. However, regardless of end effects, the variation of the potential is in general different from the variation of the entropy production.
As in other recently discussed examples, 9 an approach based on entropy production thus fails to give the fullest possible exploration of the problem. In this case it is our functional I which provides a more useful framework for analysis.
REFERENCES (Chapter II)
1. H. Busch, Stabilit1t, Labilit~t und Pendelungen in der Elektrotechnik (S. Hirzel, Leipzig, 1913).
2. R. P. Gifford and N. Page, B. S. Thesis, M. I. T. (1902). 3. H. Busch, Ann. der Phys. 64, 401 (1921).
4. W. J. Skocpol, M. R. Beasley, and M. Tinkham, J. Appl. Phys. 45, 4054 (1974).
5. M. E. Fisher and J. S. Langer, Phys. Rev. Letters 20, 665 (1968).
6. J. D. van der Waals, Z. Phys. Chem. 13, 657 (1894).
7. J. W. Cahn and J. E. Hilliard, J. Chem. Phys. 28, 258 (1958).
8. P. Glansdorff and I. Prigogine, op. cit. 9. R. Landauer, Phys. Rev. A12, 636 (1975).
III. FLUCTUATIONS A. Potentials and Distribution Functions
In the previous chapter we considered only the steady state of the ballast resistor and assumed that once it has relaxed into its steady state it remains there. But a thermodynamic system always has many states accessible to it. A full understanding of its behavior requires that we know the distribution function giving the distribution of an ensemble of
systems among the accessible states.
In equilibrium statistical physics, it was Boltzmann's H
theorem which represented the key step in relating macroscopic state functions to microscopic probability distributions. Boltzmann defined a
quantity H in terms of the probability distribution of the system. He then showed that H reaches a minimum at equilibrium, where it is equal to minus the entropy. This connected the entropy, which macroscopic studies had shown increases, with the distribution function. The proof of the H theorem may be outlined as follows:
1. All microstates of the same energy are equally likely. 2. H is a measure of the probability of any macrostate of
the system.
3. The system tends to move from less probable to more probable states.
4. H tends to decrease.
5. H is a minimum at equilibrium.
For the ballast resistor, we would like to connect our function Su, which we know decreases, with a probability distribution. Unfortunately, there is no assumption like (1) known that will enable us to prove a
the ballast resistor's probability distribution is:
P I e-4 /c (3.1)
where c is a constant, for states near a uniform steady state, and conjecture that (3. 1) remains valid everywhere. A number of
conclusions that follow from that conjecture will be found to be correct.
B. Fluctuations near the Uniform Steady State
We want to calculate the spectrum of temperature fluctuations about the ballast resistor's uniform steady state. We therefore start by assuming that a very long ballast resistor of length L is operating at a
2 2
point outside the coexistence curve in the T, i plane. With i held constant, we may expand A - i2R about the operating point:
A - i2R = aAT (3.2)
where AT(x, t) is the difference between the local temperature and the steady-state temperature. We then add a fluctuating Langevin-type heat source to the energy conservation equation (2. 4) and obtain:
OAT 09AT
c v Ot =-A + X Ox2 +
Q
(x, t) (3.3) where X is independent of temperature.We now take the spatial Fourier transform:
D 2
cv 5 AT (k, t) = -AT (k, t) - 2 T (k, t) + Q(k, t)
AT(k, t) = - FAT (k, t) +
Q(k,t)
(3.4)v where
F - (O + Xk2) (3.5) v
At first glance, this is identical to the Langevin equation. It differs, however, by reversing the roles of the force and the flow; we now have a dissipative flow opposing a random force rather than a dissipative force opposing a random flow. We will see shortly, however, that the
fluctuation-dissipation relation still holds in this case, even though the original proof of the theorem does not apply because temperature
differences and heat flows cannot be immediately expressed in terms of a perturbing Hamiltonian. 2
First, however, we will do a time Fourier transform:
(-iw+ F) AT (k, W)
-
Cv Q(k, W) or, taking the square of the absolute value of each side:2 1 1( w (.6
A T2(k, W) = 2 2 Q (k, W) (3.6)
cv (F + w)
It now remains to determine the spectrum of
Q.
There are threeindependent fluctuating sources of heat in the wire: heat transfer between the wire and the surrounding gas QA, heat conduction along the wire
Q
and fluctuations in electrical power dissipation due to Johnson noise in the wire Q 2 . We treat each of these independently:
i R
Q =QA + Q + Q.2
Q
(k, W) = Q A(k, W) + (k, W) + 2 Nk W) (3.7)We consider first Q 2 * We may expand the noise voltage,
i R
which may be a function of the applied current, about equilibrium as follows:
(V2 ( 2 )(0) + , (V2 ) (1) + 2 (V2 ) (2) +
n n n
3
Bernard and Callen compare this with the expansion of dc voltage in terms of the current:
V = iR + i2R(2) + i3R +
They show that there is a general relation (for h W << kBT):
T (V2)(n) a k T R(n+1)
2 n B
Since our current is small enough to give a constant resistance at
constant temperature, inR(n+1) << R , and therefore the higher terms in the expansion of (V2 ) may also be neglected. We are left with the
n equilibrium fluctuations:
2 2
(V) = WkBTRL
These are the voltage fluctuations, but we are interested in the power fluctuations. To calculate what we want, we must first consider the fluctuations in electric field at any point, En (x W). Now consider the integral of En over any distance I which is short, but much longer than the intermolecular spacing. This is simply the noise voltage across a resistance R2. We therefore have:
E (x', W) dx')2) =- k TRA
n ITB
x+
where E dx is a Gaussian random variable. A theorem of n
4
Chandrasekhar states that if (3. 8) is true, then V n(k, W) is a Gaussian random variable with:
(V (k, W)) 2 kBTR (3. 9)
L/2
where V n(k, W) is defined by V n(k W) = E(x, W) eikx dx. Now let us
Al ~2 ~-.
calculate Q 2 (k, W). Since the mean power delivered is i R,
Q
2Spower 2 i R
the difference between the instantaneous power delivered and i R. will take the limit L -+
Q
2R(k,w)L/2
dx e-ikx dt e-iWt E2 x, t)
1
S
dx e-ikxSdt
where E is the mean electric field.0 (E0 << E n 0 Q.2 (k, W) (Q1 2 (k, w)) 1 R e~iwt [2E E (x, t)+ E (x, t)]o n n 2
En may be neglected because 2E R 0 Vn(k, W) 4E2 = 0 (V (k, W)) R 4E2 2 kBTR = 2 IT B 8 8 Ak T 7T B 'x+ x (3.8) is (We (3. 10) (3. 11) 2 (Q 2(k, W)) 1i R
2 ' We have used the fact that at the steady state A - i2R = 0 to put
Q
2i R into a form appropriate for comparison with QA A and R are here the heat flow and resistance at the mean temperature.
We will now calculate the remaining term
QA
+ . It should be noted thatQ
is defined as a heat source, and therefore has different units from the heat flow W along the wire. Rather,Q
has the units of dW/dx.Now consider a single mode of wave vector k of the wire. This mode will satisfy a Langevin equation of the form (near equilibrium):
d
c AT(k,t) = -f AT(k,t) + Q (k, t) (3.12) where
DA 2~
f =A T k and Q =Q+
We can then conclude directly from the generalized fluctuation dissipation relation5 that:
Q(k, W)) = kBT 5+ Xk (3.13)
In order to apply the generalized fluctuation-dissipation theorem, however, we must show that we are in a region where the relationship between heat flows and temperature gradients is linear. This may be assumed for conduction within the wire. However, for the flow A, this requires more careful examination.
It is clear, first of all, that the onset of convection in the gas surrounding the ballast wire will cause some kind of singularity in the heat flow. One might hope that laminar convection would, in any case, not add additional fluctuations because it is approximately linear in the
driving force. But beyond a second singularity marking the transition to turbulence, even this frail hope is gone. Busch, however, finds that
convection does not play a role for gas pressures below about 250 mm. 6 Radiative heat transfer is also nonlinear for large temperature differences between the heat reservoir and the wire temperature
((T - T0 )/T 1). Busch shows that for gas pressures above a few mm, the radiative heat transfer is negligible compared to the thermal
conduction.6
We are left with the conduction. At first glance it might seem as though we are still far from linearity, since A is proportional to
T3 /2 - T3/2. But we recall that this is a consequence of the
tempera-0
ture dependence of the thermal conductivity of the gas. Locally, the heat flow is proportional to the temperature gradient in the gas. This will be true as long as the radius of the wire is much longer than the mean free path of the gas molecules. For Busch's wire with diameter
0. 05 mm, this required p
>
50 mm. 6We therefore find that for a wire of diameter 0. 05 mm, there is a regime of pure, linear conduction between about 50 mm and about 250 mm pressure. We can now write our entire Langevin equation
explicitly using (3. 11) and (3. 13):
(AT 2(k, W)) =c (F2 + ) kBT2 4- + + k2 c A (A 2 2 k T 4- + -- + X k (A T2 (k, W)) = B T OT (3. 14) (a + Xk2)2 + W2 c2 v
inverse Fourier transform with respect to frequency: (AT(k, t)AT(k, t+T)) (AT(k, t)AT(k, t+r)) k T= B 2 l c k T2 v Tc V 4A + + Xk2 T OT a + Xk2 4 A + O+ Xk2 T T a + k2 e 2 cv iWT a+Xk Wc 2 + + 2) + Xk - (a+Xk )II c V
We now transform with respect to wave number:
kBT 2 CO -c A+2 +R Ol + X k2 c x e k T2 Tc v + vy x -0
S
-ep-{fI
2 X+ Iri2' k2 exp - (XCk v - i~k + a vIT I
(XT I v + dk 2ff X IT - 2 k-ti( c) v T 2 k c vITI
- i~k + ' v 4 26R dk 21T a + Xk2 dW 2 5 (3.15) a 2 + Xk2 (A T(x, t) AT(x + , t + T ))k T2 cvT kBT2 + Vc
v
4( T V c I STV~r)
2 e cv~r
(Xk2 4 xcv dk 21 .2 OR) +OT exp-_c
(3. 16)
The first term is essentially the equilibrium temperature fluctuations. The second term, which is more interesting, cannot be integrated. However, the integral can be performed for 4 = 0.
(AT(x, t) AT(x, t+T)) kBT2 I c v 4- + T ~c k e -o + Xk 126 -67 / c dk 2T + equilibrium term k BT 22 c 4 +i2 -R T O
a--r
c e 2 (I e v erfc + equilibrium term k T2 = c v (OtXc) 4A + .2 bR' erfc I kBT2 + B (it c v x I')da nI
exp c v 2 c 4 x V (3. 17) vI
(c
(v Ot + X k2 (AT(x, 0) A T(x + 4, t+,r)) rj+ ik - " ITI) (A T(x, t) A T(x, t + T))The integral is found in Gradshteyn and Ryzhik.
Thus we have two terms in the correlation function. The first term has an amplitude equal to the amplitude of the fluctuations at
equilibrium in the limit of short time delay T. (The fact that these fluctuations become infinite for T = 0 should not surprise us because the average instantaneous temperature fluctuation of a body is inversely proportional to the specific heat, and the specific heat of a point is 0.) The second term represents the additional fluctuations due to the fact that the system is being driven.
Now we shall calculate the frequency-dependent spatial correlation.
2k T2 ( . e1k (A T(x, W) AT(x+j,W)) = (4 A+i 2 OR 2T ST T (a + Xk 2) 2+ c2 v (3. 18) SBT2 a+Xk 2 i4k dk + ~~(Ci + Xk 2 ) 2 +
2
-ce2 1 v a t i(A C IThe integrands have simple poles at k = + i V)2. We will assume j > 0 and after the calculation use the symmetry of the correlation
function to make the substitution ( -
I1.
The poles in the upper half C t iW c plane are at k = i : (A T(x,W) AT(x+,W)) = B(
4A
+ i2 OR, T T 1T )rr U+iWc-iwc } exp ( x ) exp -c( ( )*i 2 , ic Wc( )E2 X Wc ( o&+iitc
Ia
- iW ciWc, exp V)>iW expKK
- ic c+ I.i) 2 w:L
2 k V (a + iW vc ) 2 ,a ova -W i v
This is pure real, as it should be, since inside each pair of square brackets is the difference of two complex conjugates.
(~T(x.~) ~T(x ~,~)) = .1 )c e (a - iWc )* V -1+ k TB 2 WWX c v
a + iWc
kBT2 - V)JB
T Fe X i T L- (a + iWC ) a+iwc } -| (() e (+ iocvx
- iWC e-i( c)} (a - imc V); or, 2k T2 4A + 2 bR(AT(x,W) AT(x+4,W)) = B Re [M(W)] - T Im[M(W)]
(3 9 (3.19) where M(w) =
e
(ca+iwc) V (3.20) C. Critical FluctuationsAs equilibrium systems approach a critical point, their fluctuations slow down, become correlated over a larger distance, and grow in
amplitude. The same effects may be observed in the fluctuations we have just calculated for the ballast resistor.
Equation (3. 17) gives us the time dependence of temperature fluctuations at any point. There are two time constants, k /cv and
.+2 kB T 2
.A .2 6R i T4 + i - )!
c v/a. As one moves toward the critical point, c and X are constant, while a vanishes. Near the critical point, for the long-time variations
1
we are interested in, dependence of one term of (3. 17) on (>T)~ 2 will not
contribute significantly to the time dependence of the correlation function. The correlation time will thus be essentially cv/a; however, this time does not have the precisely defined meaning of the correlation time of a purely exponential decay.
Since near the critical point slow fluctuations become dominant, we may obtain the behavior of the correlation range by taking the
low-frequency limit of (3. 19). This is a complicated function, which will display oscillations for large J and small a and W. However, if we average over these oscillations, the correlation function depends on distance only through a scale distance (/a
),
which is multiplied by a numerical factor of order 1.Finally, the amplitude of the fluctuations diverges. We are not interested in the infinity which we find when the temperature fluctuations of a single point at equilibrium are calculated. To avoid this difficulty, we will consider (3. 17) in the limit of time delay r which is very short, but still long enough that the second term dominates. In this case, we find the mean aquare amplitude going as a~S.
To summarize, we have three critical exponents for the fluctuations' correlation time, correlation length, and amplitude:
Tcorrelation = a (3. 21a)
Scorrelation ai (3. 21b)
(3. 21c)
If A - i2R is analytic at the critical point, then if the wire is held at the critical temperature while the ambient temperature is raised, 0 will be proportional to the distance from the critical point C, and the exponents of (3. 21) will be the critical exponents usually defined.
For the validity of this theory, it remains to be shown that the amplitude of the fluctuations calculated from it is much less than the smallest distance from the critical point attainable in an experiment.
For a typical iron or alloy wire of diameter 120 gm, we may take:6 A = 107 erg sec-cm c = 8 x 103 erg-cm sec-K X = 8 x 102 erg-cm sec- K T . i 103 oK crit .2 OR A OT T
If we take, optimistically, , = 10-3 A/T, we obtain from the second term of (3. 17) at T = 0,
(AT2) = 1. 7 x 10-11 (OK)2
and the rms temperature fluctuation is about 4 microdegrees, far less than the temperature control attainable in practice for the system. D. Potential of Fluctuations
In Eq. (3. 14), we have found the spectrum of temperature fluctua-tions in each spatial mode of the system. Since AT(k) is related to the
Gaussian random process Q(k) by the linear equation (3. 4), it is also a Gaussian random process. Therefore its first two moments are
sufficient to specify its entire distribution function as follows:8
P[AT(k)] = N exp AT2 W (3. 22)
2 (AT (k, t))
From (3.15) we have:
k T2 A + + X 2
(AT 2(k, t)) = Orc T (3.23)
The third term in the numerator will be negligible whenever k2 << dA Now dA/dT is the maximum value of a. As we showed in the last section, (.U) is the correlation length of the system. We will not be interested in those fluctuations whose length is shorter than the smallest possible correlation length of the system. We therefore drop the >Xk2 term in the numerator of (3. 23). We cannot drop the same term in the denominator because it will become significant when a becomes small near the critical point.
We now calculate the change in the potential function (2. 7) due to a fluctuation of the form AT sin kx:
L T+ AdT sin kx 2 2 2(d sinkx)2 ' A i s= dx ST(A-i2R )dT' +} AT 2 ink) oT S L A T sin kx
= L dx ATsink AT' dAT' + X2k2 A2 cos2 kx}
o o
L
= dx {X AT 2 sin2 kx + 2 k2 A2 cos2 kx
(3. 23) gives the spacial spectral density of temperature fluctuations for continuous k. In order to obtain the amplitude of a given fluctuation
L mode AT sin kx, we must divide by the density of states
2 A OA
(AT 2) = - L 2 (3.25)
cv L a+ (k.
Substituting (3. 24) and (3. 25) into the exponent of (3. 22), we arrive at a distribution:
P[AT(k)] = N exp -A2 k) (3.26)
B (A+1dA>
c \T 4dT)
v
Since A4t(k) = AT2(k), the contributions of different orthogonal modes to the total potential of the wire will simply add, and so we have the following expression for the probability of any given configuration of temperatures along the entire wire:
P{AT(x)} = N' exp
{
(AT(x)) (3.27)2 XkBT (A +1 dA)
c \T 4 dT}
This is, of course, of the same form as the Boltzmann factor of equilibrium statistical mechanics, and probability distributions of this form have been obtained in the previously studied non-equilibrium cases. 9 We shall conjecture that the probability (3. 27) applies to all temperature distributions AT(x), at any current, with the temperature in the
we have a constant
2 kBT A 1 A\
c ~ c \T +
T0
(3.28)v
The properties of the potential d derived in section II. D are all consistent with this hypothesis. Since the steady state is a local
extremum of I, it corresponds to a state more probable than any of its neighbors. If the system tends to evolve from less probable to more probable states, we would expect to find d : 0, as given by Eq. (2. 12). And in the description of Eq. (3. 27), the metastable state, as identified by the Maxwell construction (2. 3), is always less probable than the stable state.
The quantityl now plays the same dual role as the free energy in equilibrium. It arises in the solution of the mean macroscopic equations of state of the system. Yet it also appears in the probability distribution function for microscopic fluctuations.
REFERENCES (Chapter III)
1. P. Ehrenfest and T. Ehrenfest, Handbuch der Physik. 2. H. B. Callen and T. A. Welton, Phys. Rev. 83, 34 (1951).
3. W. Bernard and H. B. Callen, Rev. Mod. Phys. 31, 1017 (1959). 4. S. Chandrasekhar, Rev. Mod. Phys. 15, 1 (1943), Eq. (147). 5. L. D. Landau and E. M. Lifschitz, Fluid Mechanics (Pergamon,
London, 1959) pp. 523-529.
6. H. Busch, Ann. der Phys. 64, 401 (1921).
7. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic Press, New York, 1965), Integral
No. 3. 466. 1.
8. W. B. Davenport and W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw-Hill, New York, 1958). 9. See references 4, 7, 8, and 12 of Chapter I.
IV. LONG-RANGE ORDER
It is easily seen that for applied voltages in the two-phase region, the only stable steady state solutions of the energy conservations
equation (2. 4), corresponding to minima of the potential (2. 7), have the wire divided into a single hot region and a single cooler region. (For certain boundary conditions at the ends of the wire, there will be two regions of one of the phases. ) This is true for any length wire. In this state, the ballast resistor exhibits long-range order; knowledge of the phase at a single point gives us knowledge about the phase at a point
indefinitely far away.
But if the probability distribution of the different possible states of the system is given by (3. 27), as we have conjectured, there will be no long-range order. This may be seen from the following arguments: Since the two phases have the same potential per unit length, the total potential of the wire will be a constant plus contributions made by
inter-facial regions. If the interfaces are sufficiently far apart, they will all have the same "shape", and each will make an identical contribution AL to the potential.
Any given state with n+1 interfaces is therefore less likely than the state with one interface by the factor e-n
I/c.
But while there is always just a single state (or two states) with one interface, the number of possible states with n+1 interfaces can be made arbitrarily large by increasing the length of the wire. For a sufficiently long wire, then, it will be more likely that there are n+1 interfaces than a single interface.In equilibrium statistical mechanics, where the distribution
function always has the same form as (3. 27), it has been known for some time that one-dimensional systems cannot have long-range order (without
.1
.
long-range interactions). Landau and Lifschitz give a proof essentially the same as the proof of the previous paragraph, and show that a
sufficiently long one-dimensional system whose most likely state is one region of each phase will break up into random-length segments of alternate phase.
In order to examine the validity of this consequence of the conjecture that (3. 27) applies everywhere, we will determine by a different method, which does not involve any use of the quantity 1,
whether an infinitely long ballast resistor will exhibit long-range order. If we find that there is no long-range order, this will tend to confirm our hypothesis about the dependence of the probability distribution on O.
Consider an infinitely long ballast resistor. If in fact the ballast resistor has long-range order, it will divide into one region which is predominantly at Ta and one region which is predominantly at Tb.
Except in a boundary layer, a temperature fluctuation will always decay away quickly toward either Ta or Tb. This enables us, in the phase initially at Ta to consider only fluctuations which bring a segment of wire to temperature T Such fluctuations change the temperature of a macroscopic segment of wire by several hundred degrees, requiring an energy transfer many orders of magnitude larger than kBT. If the ballast resistor has long-range order, we would then expect the
fluctuations to be few and far between. Based on the assumption that the probability of any point being inside such a fluctuation is small, we will calculate that probability. We will find that it diverges. The probability is therefore not small. This argues strongly that the infinitely long ballast resistor does not have long-range order.
both because of fluctuations and because of forces which make such segments disappear, forces which are reflected in the gradient term of the potential (2. 7). We shall show that these two influences combine to make the length of these segments undergo a biased random walk, and that the biasing is too weak to prevent the lengths of some segments from "escaping" to infinity.
We will start by calculating from the energy conservation
equation (2. 4) the decay of a single segment of Tb in an infinite wire of Ta, ignoring fluctuations. We assume constant current, which in the limit of an infinite wire is the same as constant V. Now a finite
segment of wire of phase b will not really have a uniform temperature Tb. Its exact temperature will be given by solving Eq. (2. 4). For temperatures near Tb, where we can take X as a constant and assume A - i2R to be linear in T, Eq. (2. 4) becomes:
c(T-T d 2T (4.1)
dx2
If we measure x from the center of our segment, this may be integrated to give:2
T(x) = Tb - ke(a/XAx (4.2)
If the length A of the segment of phase b is defined to be the distance between the two points which are at some arbitrary temperature Tb we can evaluate the constant of integration:
k = Tb - T(o) = Oe
Since the temperature at the point where dT/dX = 0 is less than Tb, the Maxwell construction Eq. (2. 3) will no longer be satisfied and the
segment will shrink. To calculate the rate of shrinkage, we will assume that the phase interface moves at a uniform velocity u =
tdA/dt
withoutchanging shape. We may then use Eq. (2. 5) and find:
- e /)iA =(a y eP
(4.3)
b dT
T vd a
For the wire described by Busch, 2 taking 9 = 50 0K, the decay time of a segment in seconds is related to its length in centimeters by the
approxi-3L
mate formula t = e , so that a one-centimeter segment will last about 20 seconds and a 15-centimeter segment will last about 1010 years.
Now we must take into account the effect of fluctuations on our segment. In the case where (to first approximation) a first-order
phase transition occurs, we would expect these segments of Tb to be few, and we can then neglect the possibility of two such segments colliding, or of a single segment being broken into two by a fluctuation from Tb to Ta-The length A of a single segment of Tb will then be described by a
random process, and this process will be a Markoff process. Its probability distribution will therefore be governed by:3
dP( [D,(1) P(.)] (A > s)
(4.4) i = 1
where s is a short distance chosen to exclude segments which are too short to allow the approximation of Eq. (4. 3). Since we will be interested only in the asymptotic behavior of P(A) for large A, we can drop all terms of higher than second order. D is simply the rate of shrinkage of a segment, which we just calculated. We can therefore arrive at a Fokker-Planck equation for the length of a segment:
The steady-state solution of this equation is:
P(A) = D exp 1dA
(4.6)
+ dA' exp 2 d "} (A> s)
If the total number of fluctuations per unit length of wire is to be a finite constant, then J, which is the net rate of creation of fluctuations, must be zero. Since the shape of the transition region between phases a and b becomes independent of A for sufficiently large A, D2 should approach a
constant for large A. We conclude that P(A) approaches a non-zero constant for large A.
This implies that the average length of a fluctuation, AP(A)dA, is infinite. Since the average number of fluctuations centered in a unit length of wire must be nonzero, we conclude that the proportion of the wire occupied by fluctuations is infinite. This certainly contradicts our
assumption that that proportion is small.
Strictly speaking, we must prove not only that the proportion of phase a at Tb is not small, but that it is the same as the proportion of phase b at Tb. But our calculation of P(A) indicates that. Ignoring
downward fluctuations, we showed that the mean length of a segment of Tb in phase a is infinite. This implies that the true mean length of such a segment is simply the mean distance between two fluctuations from Tb to Ta. But this is necessarily the mean length of a segment at Tb in
phase b. By the same reasoning, segments of Ta must have the same mean length in both phases. Since any continuous region of a one-dimensional system must contain equal numbers (within + 1) of the two kinds of segments, we conclude that both phases have the same proportion of their lengths at each temperature. In other words, the phases are indistinguishable.
To determine whether this effect is of significance in practical experiments, we will calculate the mean length of a single-phase segment of the wire from the probability distribution (3. 27). According to that equation, the probability of having n interfaces between phases is
S N1 2
F
T1 -enAX/ c(47 (n) = N" 2LnTn)!
j ' (4)where T is the total number of possible sites for interfaces. T is evidently equal to the length of the wire L divided by some scale length h which indicates the smallest scale on which different states are to be distinguished.
The part of Eq. (4. 7) in brackets is the number of different possible combinations of locations for the interfaces. Since hot and cold phases must alternate, we specify the phase of each region for a given set of interface locations by specifying the leftwardmost phase, which may be either hot or cold. Thus, two times the quantity in brackets is the total number of states with n interfaces. N" e is, as was derived earlier in this chapter from Eq. (3. 27), the
probability of a given state of n interfaces. The constant c is defined by Eq. (3.28).
few in number, T will be much larger than n for values of n which contribute significantly. We may thus approximate
2 N" Tn e-nAt/c
(n) n!
Since this is a Poisson distribution, the mean value of n is given by the well-known formula as:
(n) = T-A$/c _ L e-Al/c (4.8)
The mean length of a segment is
L h e (4.9)
We shall now estimate the numerical value of Eq. (4. 9). As was mentioned in section II. C, the two terms of the potential (2. 7) are equal at the steady state. We are very close to the steady state, and in
Eq. (4. 3) we already approximated the form of the interface by the steady state form. We will therefore estimate the potential by
estimating the second term of (2. 7) and doubling. This may be done by approximating the mean value of dT/dx by the temperature difference
Tb - Ta divided by the interface thickness d. We then have:
bx a
AZ= 2 Tb dx4)t(X (Tb -)Ta
a
bA A
If we further approximate S T 2 , Eq. (3. 28) gives: XkB TA c B Then 2 X cv b - Ta)2 el o/c de k
If we use the same numerical values as in section III. C, and take T -T = 5000K, T = 10000K, and d = 1 cm, we get:
b a
ea/ C (1018) (4.10)
By Eq. (4. 8), this confirms our assumption that T >> n.
Whether we choose h to be equal to d or some other length does not really matter. Any length h we might choose, multiplied by e(101 8 gives a very long mean segment length (4. 9).
We can therefore conclude that the mean length of a single-phase region is so long that the system's lack of long-range order is of no possible experimental significance whatsoever.
REFERENCES (Chapter IV)
1. L. D. Landau and E. M. Lifschitz, Statistical Physics, (Addison Wesley, New York, 1969), Section 152. 2. H. Busch, Ann. der Phys. 64, 401 (1921).
3. R. Stratonovich, Topics in the Theory of Random Noise, (Gordon and Breach, New York, 1963), Vol. 1, p. 58. 4. M. Lax, Rev. Mod. Phys. 38, 359 (1966).
V. DYNAMIC RESPONSE IN THE TWO-PHASE REGION
A. Response Time
The technological application of the ballast resistor is as a device which passes a constant current when a varying voltage is applied to it. Like any other filter, its response is frequency-dependent. Voltage variations of sufficiently high frequency, added to the constant driving voltage, will cause current variations. It is therefore of practical as
well as theoretical interest to find the frequency response of the ballast resistor.
For ease of calculation, we will examine the system's transient response to an instantaneous change in the applied voltage. This
transient response is, of course, the Fourier transform of the transfer function giving the frequency response.
Consider a very long wire of length L. The total voltage drop across the wire is given by:
L
V = i R(T) dx (5.1)
Assume that at the steady state with V = V0, the phase at Ta has length 1a and the phase at Tb has length Ab. In the remainder of this section, Tao and Tbo will designate the temperatures which satisfy the Maxwell
construction Eq. (2. 3), and Ta and Tb will designate the instantaneous temperatures of phases a and b, which do not necessarily satisfy (2. 3).
(T o> T a). (bo Tao'
Suppose that we instantaneously make a small change in the applied voltage. We will assume that the response of the system has two distinct stages. In the first stage, the temperature of each single phase region
