Bispectrum Theory for Brownian Motion of
Electrical Charge in Nonlinear RC-circuit
Boris M.Grafov
A.N.Frumkin Institute of Electrochemistry of Russian Academy of Sciences
31 Leninskii prospekt, Moscow 119071, Russia
Abstract. The purpose of present report is to provide the bispectrum theory for Brownian
motion of charge in the equilibrium non-linear RC-circuit (both resistance R and capacity C
are nonlinear). The theory is based on the dual Langevin linear equations and on the
Stratonovich fluctuation-dissipation relation for the voltage noise bispectrum of resistor.
The equation for the asymmetry of the equilibrium electrical charge fluctuations in the
nonlinear capacity is derived. This equation differs essentially from the Gibbs
thermodynamics equation in several aspects. In line with the bispectrum theory, the
asymmetry of equilibrium charge fluctuations depends on the resistance nonlinearity and is
independent of the capacitance nonlinearity. We arrive at the fundamental conclusion that
the Gibbs statistical method is inapplicable to the description of nonlinear thermodynamic
fl
i
INTRODUCTION
There are two general theoretical ways to analyze the thermodynamic
fluctuations. The classic way is to use the Gibbs statistical method. The second
one is the noise approach. Let consider the charge fluctuations in the capacity. In
line with noise approach the variance < q 2 > and asymmetry (skewnees) < q 3 >
of thermodynamic charge fluctuations can be found by integrating single
spectrum < qω qω* > and bispectrum < qω qν qω* + ν > over all frequencies ω and ν :
∞
1
< q >=
dω < qω qω* >
2π −∫∞
2
< q3 > =
1
(2 π) 2
∞
∞
−∞
−∞
∫ dω ∫ d ν < qω qνqω* + ν >
(1)
(2)
It is well known that, in the case of Brownian movement of electrical charge in
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71
the equilibrium RC-circuit, both ways lead to the same expression for the charge
variance < q 2 > [1]:
< q 2 > = kTC
(3)
where k is Boltzmann’s constant, T is the temperature, C is the capacity. For
asymmetry of charge fluctuations in capacity the Gibbs statistical method yields
[1]:
< q 3 > = ( kT ) 2 dC / dE
(4)
where dC / dE is the derivative of capacity with respect to the voltage.
The purpose of this paper is to consider the charge fluctuations asymmetry in
the nonlinear RC-circuit (Fig.1) from the viewpoint of noise theory.
R
C
ε(t)
Fig.1. The RC-circuit: ε(t) is the Thevenin voltage noise generator
NYQUIST’S THEOREM
It is well known that the Nyquist fluctuation-dissipation theorem is dual and
given by equations [1]:
< ε *ω ε ω > = 4 kT Re Zω
(5)
< iω* iω > = 4 kT Re Gω .
We use the following notations: Zω is a small-signal impedance; Gω = Z
(6)
−1
ω
is a
small-signal admittance; < ε ε ω > is a single spectrum of voltage noise ε( t ) ;
*
ω
< iω* iω > is a single spectrum of current noise i (t ) ; t is time. The voltage noise
ε( t ) characterizes the Thevenin noise circuit. The current noise i (t )
characterizes the Norton noise circuit.
DUAL LANGEVIN EQUATIONS
The Nyquist fluctuation-dissipation theorem (5)-(6) is applicable both to linear
and nonlinear equilibrium systems [1-2]. We have used [3] this property of the
Nyquist theorem to generalize the Langevin stochastic equation to the nonlinear
Brownian movement in the form of dual stochastic equations:
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∞
ε( t ) =
∫ dt H
( t − t1 )i ( t1 )
(7)
(t − t1 ) ε (t1 ) ,
(8)
E
1
−∞
∞
i (t ) =
∫ dt H
I
1
−∞
where H E ( t ) is a system function characterizing the small-signal voltage
response and H I (t ) is a system function characterizing the small-signal current
response. The Langevin stochastic equation (7) is the map from the Norton
current noise to the Tevenin voltage noise. The Langevin stochastic equation (8)
is the map from the Tevenin voltage noise to the Norton current noise.
BISPECTRUM OF CURRENT FLUCTUATIONS
The dual Langevin stochastic equations (7)-(8) can be considered as the linear
filter equations. One can conclude on the basis of (8) that bispectrum for current
i ( t ) of the circuit in Fig.1 is
< iωiνiω* + ν > = Gω GνGω* + ν < ε ω ε ν ε *ω + ν >
(9)
where the star denotes the complex conjugate quantity. Admittance Gω of
circuit in Fig.1 is given by equation:
1
(10)
Gω =
R + jωC
where j is the imaginary unit, R is the small signal resistance, and C is the
small signal capacitance. Fluctuating current i ( t ) and fluctuating charge q ( t )
are interrelated by equation:
i ( t ) = dq ( t ) / dt .
(11)
Equation (11) is linear. Therefore, we can use the following equation of linear
filter theory:
< iωiνiω* + ν > = ( jω )( jν)( − jω − jν) < qω qν qω* + ν > .
(12)
It is known [4] that the bicovariance function of nonlinear resistor coincides
with that for the nonlinear white noise. The corresponding voltage bispectrum is:
< ε ω ε ν ε *ω + ν > = B( kT ) 2 dR( I ) / dI
I =0
(13)
where B is the number and dR( I ) / dI I = 0 is the derivative of small signal
resistance R( I ) with respect to steady-state current I at the equilibrium point.
According to Stratonovich [4]
B = 24 .
(14)
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CHARGE FLUCTUATION ASYMMETRY
Combining (9),(12),(13), we obtain for the charge fluctuations bispectrum:
B( kT ) 2 dR( I ) / dI I = 0
< qω qνqω* + ν > =
(15)
( jω )( jν)( − jω − jν)( R + jωC )( R + jνC )( R − jωC − jνC )
After substituting (15) for < qω q νqω* + ν > in (2) and integrating over both
frequencies ω and ν , we obtain for the charge fluctuation asymmetry the
following expression:
B
( kT ) 2 CR − 2 dR( I ) / dI I = 0 .
(16)
< q3 > =
12
It is seen that equations (4) and (16) are strongly different. According to the
Gibbs statistical method, the charge fluctuation asymmetry is independent of
linear or nonlinear resistor properties and is absent for the linear capacity.
According to the noise approach, the charge fluctuation asymmetry is
independent of nonlinear capacity properties and is absent for the linear resistor
only. We arrive at the fundamental conclusion. The Gibbs statistical method
does not work to calculate the asymmetry and higher cumulants of
thermodynamic fluctuations. Therefore, a tremendous task arises - to create the
statistical kinetics that could describe the nonlinear thermodynamic fluctuations.
ACKNOWLEDGMENTS
Author is grateful to L.B.Kish, A.M.Kuznetsov, S.F.Timashev, and
R.M.Yulimetiev for useful discussions of nonlinear noise problems.
This work was supported by the Russian Foundation for Basic Research,
project no. 02-03-32114.
REFERENCES
1. W.Bernard and H.B.Callen, Rev. Mod. Phys. 31, 1017-1044 (1959).
2. Sh.Kogan, Electronic Noise and Fluctuations in Solids, Cambridge University Press, New
York, 1996, pp.57-62.
3. B.M.Grafov, Fluctuation and Noise Letters 4, L617-L622 (2004)
4. R.L.Stratonovich, Izv. VUZ Radiofizika 13, 1512-1522 (1970)
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