A “thermodynamical” proof of the extended steady-state Nyquist theorem

Physica

46 (1970) 605-608

0 North-Holland

Publishing

Co., Amsterdam

A “THERMODYNAMICAL” OF THE

EXTENDED

PROOF

STEADY-STATE

NYQUIST

THEOREM

V. SERGIESCU Institutul

de Fizica

al Academiei,

Received

Bucuresti,

Romania

9 June 1969

Synopsis The

thermodynamical

supplemented

proof

of the

by the information

theory

of the noise in the non-equilibrium fluctuation-dissipation

theorem

Nyquist

formula

for the equilibrium

in order to derive

the frequency

state

is

dependence

steady state. The result agrees with the extended

of Landsberg

and Cole (1967) and Lax

(1960).

According to the Nyquist theorem the noise spectra of the current i, voltage u, and power fi of an impedance Z(W) at thermodynamical equilibrium are, respectively, G(i; v) = 4kT.We

1 ~. Z(m) ’

G(u; v) = 4kT *Se Z(w) ;

G(fi; v) = kT,

(1)

where T is the temperature of the system and cc)= 27~. On the other hand, recent extensions of the fluctuation-dissipation theorem (Landsberg and Cole r), Laxs)), of which the Nyquist theorem is but a particular to following formulae under general steady-state conditions : G,(i;v)

= 4kT.C,*9e--

case, lead

1 Z,(W) ’

Gs(u; v) = 4kT.Cs*B?e

Z,(m) ;

G,(fi; v) = C,.kT.

(2)

Here C, is a frequency-independent correction factor determined by the response of the system to a constant external force. The index I’s” appearing in eqs. (2) shows that, apart from C 5, the principal difference between the equilibrium and the steady-state formulae is the substitution of Z,(W) (the impedance for nonzero average current, i.e. in the neighbourhood of the steady-state) for Z(o) (th e impedance for zero average current, i.e. in the neighbourhood of thermodynamical equilibrium). A common assumption to eqs. (1) and (2) is the validity of the linear noise approximation. 605

606

V. SERGIESCU

Fig. 1. Modified Nyquist

circuit for steady-state

noise.

The aim of the present note is to show that it is possible to derive eqs. (2) essentially by the argument used initially by Nyquist in deriving eqs. (l), supplemented by the information theory (Jaynesa), Katz*)). In line with the thermodynamical character of the derivation the constant Cs remains undetermined. However, the frequency dependence given by Zs(o) results as an universal law valid for all systems. Since any one of the three equations (2) results from the other two by conventional circuit arguments, we derive only the simplest one, that is the expression of G,(+; v). With this in mind, we consider the circuit examined by Nyquist (fig. l), modified in order to permit the establishment of a steady-state current through the impedances Z,(o), Z:(w) owing to the voltage sources B. The capacitors are introduced in order to keep this current out of the transmission line .A’, which is the thermodynamical system of interest in the reasoning. There are no reflections at the ends of the line Z because its characteristic impedance is supposed equal to Zg(~). Now wait till the steady state is established and then isolate the system C by disconnecting the impedances Z,(w) and Z:(w). Immediately after isolation the system is not in equilibrium but it will evolve towards the equilibrium state corresponding to the new external conditions. Nyquist’s proof (Nyquists), Pfeifere)) used the fact that progressive waves travel along the line in both senses, the electrical noise power emitted by the impedance Zs(~) in the frequency interval (v, Y + dv) being carried away by the waves of the same frequency travelling from the left to the right. This leads finally to the equation G($; v) dv = dv,

(3)

where is the average energy of a stationary wave of frequency v (i.e. of the sum of the two corresponding progressive waves) in the stationary regime reached before isolation. Since this regime is not an equilibrium one, is rtot equal to kT and we do not obtain the result (1). It is here that the information theory shows its usefulness. The problem consists in calculating the distribution function for the energies of the various degrees of freedom. The system is completely characterized by these energies (the electric and magnetic energies are to be counted separately) ; let P($‘B, ZO$,ZP.$~,u.$‘, . . .) be the probability distribution function, @‘g and w;’ the magnetic and electric energies of a stationary wave 1, BY?~ and wi’ the magnetic and electric energies of another

THERMODYNAMICAL

PROOF

OF EXTENDED

NYQUIST

THEOREM

607

stationary wave 2, and so on. According to the information theory P is to be determined by solving the following conditional extremum problem: J P *In P . dwyg dwf dwrg dwf . . . . = maximum,

S z -k

.dw”,‘.dwFg.dw;“.

S P.dwyg (W)

=

+ (w$

... =

+ + +.a*

= s w~%.P.d+g.d@. Applying

the

conventional

1, =

. . . + ~w~~.P.&@~-d@‘Lagrange

multipliers

method,

(4) . ..+...=const. this gives

the

equation lnP+

1 +A+~*~(w~p+w~l)=O i

and therefore

P(wfp", w;‘, wy,

wf,...)

= const’.exp[-p*x

i

(w%f”+ wz’)],

(5)

where A, p are Lagrange multipliers. It follows that the values of , = (wyg> + are independent of the index “i”, that is, in the one-dimensional case, of the frequency Y. This milder version of the equipartition theorem, which asserts the equipartition without giving the precise value of , suffices to justify the frequency independence of the noise spectrum (3), leading finally to eqs. (2). The correction factor CS could be determined by finding out the Lagrange multipliers 1, y, but this would involve more specific statistical arguments going beyond the original frame of the Nyquist reasoning. In conclusion, the extended Nyquist theorem (2) involves, firdy, the multiplication by a frequency-independent (but possibly temperaturedependent) factor C, which is determined by the steady-state response to a constant external force and, secondly, the substitution of the steady-state impedance Z,(W) for the equilibrium-state impedance Z(W). Z,(w) and Z(o) characterize the quasi-steady-state response of the system to a variable (harmonic) external force superposed on a steady state and equilibrium state, respectively. It is the substitution Z(W) + Z,(w) which results from the information-theoretical extension of the original thermodynamical proof of Nyquist. This substitution is equivalent to the frequency independence of the noise density of the electrical power emitted by the system in the steady state.

REFERENCES

1) Landsberg, P. T. and Cole, E. A., Physica 37 (1967) 309. 2)

Lax, M., Rev. mod. Phys. 32 (1960) 25.

608

THERMODYNAMICAL E. T., in Statistical

PROOF

3)

Jaynes,

4)

(New York, 1963) p. 181. Katz, A., Principles of Statistical W. H. Freeman

5)

Nyquist,

6)

Pfeifer,

Physics,

OF EXTENDED

1962 Brandeis Mechanics

and Co. (San Francisco,

H., Phys.

Rev.

H., Elektronisches

32 (1928) Rauschen

NYQUIST

Lectures,

THEOREM

Vol. 3, W. A. Benjamin

- The Information

Theory

Approach,

1967).

110. I, B. G. Teubner

(Leipzig,

1959) p. 34.