Fluctuations of Entropy Production in Partially Masked Electric Circuits: Theoretical Analysis

0.11 MB·

by  K. -H. Chiang

Fluctuations of Entropy Production in Partially Masked Electric Circuits: Theoretical
Analysis
K.-H. Chiang, C.-W. Chou, C.-L. Lee*, P.-Y. Lai, and Y.-F. Chen*
Department of Physics, National Central University, Chungli, Taiwan
In this work we perform theoretical analysis about a coupled RC circuit with constant driven
currents. Starting from stochastic differential equations, where voltages are subject to thermal
6 noises, we derive time-correlation functions, steady-state distributions and transition probabilities
1 of the system. The validity of the fluctuation theorem (FT)[1, 2] is examined for scenarios with
0 complete and incomplete descriptions.
2

n
a I. THEORETICAL STUDY: THE COMPLETE DESCRIPTION
J

8 We consider the coupled RC circuit as shown in Fig. 1. Two RC circuits of resistances and capacitances (R ,C )
1 1
2 and (R ,C ), respectively, are coupled through a third capacitance C . The two RC circuits are subject to constant
2 2 c

] driven currents I1 and I2, and the voltage differences acrossthe resistors are denoted as V1 and V2, respectively. The
h equation of state of this circuit is
c
e MˆV~˙ +V~ −ξ~=V~ , (1)
d
m
V ξ I R R (C +C ) −R C
t- where V~ ≡ V12 ,ξ~ ≡ ξ12 ,V~d ≡ I12R12 , and Mˆ ≡ 1−R12Cc c R2(C21+cCc) . The two resistors are
a (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
thermalizedattemperatureT,whilevoltagesacrosstheresistorsfluctuateduetotheJohnson-Nyquist(thermal)noises
t
s ξ andξ . ThenoisesareassumedtobeuncorrelatedandGaussianwhite,andtheysatisfythefluctuation-dissipation
. 1 2
t relation
a
m hξ~(s)ξ~T(s′)i=Γˆδ(s−s′), Γˆ ≡ Γ1 0 , (2)
0 Γ
- 2
d (cid:18) (cid:19)
with Γ ≡2R k T for m=1,2, where k is Boltzmann’s constant.
n m m B B
o
c
[

2
v
6
7
7
4
0
.
1
0
6 FIG. 1. Illustration of thecoupled RCcircuit.
1
:
v Via the change of variables V~′ ≡ V~ −V~ , the equation of state can be rewritten as MˆV~˙′ +V~′ = ξ~, which is
i d
X mathematically identical to that of a non-driven circuit. The solution of the stochastic equation is
r t
a V~′(t)=e−Mˆ−1tV~′(0)+ dsMˆ −1e−Mˆ−1(t−s)ξ~(s). (3)
Zs=0
In our work we only focus on the steady-state condition, where the first term in Eq. 3 damps out.
UsingEq.2andthefactthatΓˆ−1Mˆ issymmetric,time-correlationfunctionsbetweenvoltagesignalscanbederived
as
hV~′(t′)V~′T(t+t′)i
t′
= dsMˆ−1e−Mˆ−1(t′−s)Γˆe−Mˆ−1,T(t+t′−s)Mˆ−1,T
Z0
∞
= dsMˆ−2e−Mˆ−1(2t′−2s+t)Γˆ −→ 1Mˆ−1e−Mˆ−1tΓˆ, (4)
Z0 t′→∞ 2
2
and variances and covariances are just their special cases when t→0. Using the method of diagonalization, we then
derive
A e /λ +A e /λ
′ ′ ′ ′ 1 1 1 2 2 2
hV (t)V (t +t)i=
1 1 2(λ −λ )2
2 1
A e (λ −M )/λ +A e (λ −M )/λ
′ ′ ′ ′ ′ ′ ′ ′ 1 1 1 11 1 2 2 2 11 2
hV (t)V (t +t)i=hV (t)V (t +t)i=
1 2 2 1 2M (λ −λ )2
12 2 1
A e (λ −M )2/λ +A e (λ −M )2/λ
′ ′ ′ ′ 1 1 1 11 1 2 2 2 11 2
hV (t)V (t +t)i= , (5)
2 2 2M2 (λ −λ )2
12 2 1
whereλ1andλ2aretheeigenvaluesofMˆ (λ1isassignedasthelargerone),em ≡e−t/λm,A1 ≡Γ1(λ2−M11)2+Γ2M122,
A ≡Γ (λ −M )2+Γ M2 , andM arethe (m,n) element of the matrix Mˆ. Moreover,the correlationsbetween
2 1 1 11 2 12 mn
V~ and ξ~can be derived as hV~′(t′)ξ~T(t′+t)i=0 (no causality) and hξ~(t′)V~′T(t′+t)i=2hV~′(t′)V~′T(t+t′)i.
The Fokker-Planckequation of the full circuit can be shown to be
1
∂P(V~,t)/∂t=∇·[Mˆ −1V~′P(V~,t)]+ ∇·Mˆ−1Γˆ(Mˆ−1)T∇P(V~,t), (6)
2
and the steady-state distribution of V~ is
P (V~)∼exp(−V~′TΓˆ−1MˆV~′). (7)
ss
Forconvenienceweusethesymbol“∼”todenotethattheequalityholdsuptosomenormalizingconstantthatremains
invariantinthetime-reversalprocess. Notethatinthenon-drivencasetheexpressionreducestotheBoltzmannfactor
of the stored energy in capacitors.
The transition probability of the complete description, under an infinitesimal change in time, can be derived as
P (V~(t+dt)|V~(t))∼e−dtξ~TΓˆ−1ξ~/2 =exp[−dt(MˆV~˙ +V~′)TΓˆ−1(MˆV~˙ +V~′)/2], (8)
F
where V~˙ ≡ [V~(t +dt)−V~(t)]/dt, and ξ~ represents the required noises for such transition. In the corresponding
time-reversalprocess,
P (V~(t)|V~(t+dt))∼exp[−dt(−MˆV~˙ +V~′′)TΓˆ−1(−MˆV~˙ +V~′′)/2], (9)
R
as one replaces V~˙ with −V~˙ and V~′ with V~′′ =V~ +V~ , the latter resulting from inversion of driven currents. Subse-
d
quently, the net dissipation can be shown to be
P (V~(t+dt)|V~(t))
F
dS =k ln
Q B P (V~(t)|V~(t+dt))
R
=−2k dtV~TΓˆ−1Mˆ V~˙ +2k dtV~TΓˆ−1V~
B B d
=dtV~ ·~i /T, (10)
R
where~i represents the currents through the two resistors. The corresponding change in Shannon entropy is
R
P (V~(t+dt)) dlnP (V~(t))
ss ss
dS =−k ln =−k dt
Sh B Pss(V~(t)) B " dt #
=2k dtV~′TΓˆ−1Mˆ V~˙ (11)
B
and the total entropy change dS =dS +dS is
tot Q Sh
dS =−2k dtV~ TΓˆ−1MˆV~˙ +2k dtV~TΓˆ−1V~
tot B d B d
=dt(2V~′·I~ −ξ~·I~ +V~ ·I~)/T . (12)
d d d d
3
For our current discussions, V~ and I~ are constants, and therefore dS is a Gaussian random variable, while
d d tot
hdS i=dtV~ ·I~/T =dt(I2R +I2R )/T. One can show that
tot d d 1 1 2 2
hdS2 i−hdS i2 =(dt)2h(2V~′·I~ −ξ~·I~)2i/T2
tot tot d d
=2k dtV~ ·I~/T =2k hdS i. (13)
B d d B tot
It is straightforward 1 to demonstrate that FT holds for any Gaussian random variable whose ratio of variance over
meanvalueis 2k . Moreover,withthe aidoftime-correlationfunctions,one canalsodemonstratethe validity ofFT
B
over finite-time processes, where ∆S = τ dS , and h∆S i=τV~ ·I~/T.
tot,τ t=0 tot tot,τ d d
R
II. REDUCED DESCRIPTION (A): NAIVE DESCRIPTION
In the reduced descriptions, we neglect the signal V intentionally, and we would like to check whether FT can
2
still validate with the knowledge of V only. Note that since the current throught R is not measured, the actual
1 1
dissipationthroughtheresistorisnotknown,whilethereexistmanymethodstowardsguessinganeffectivedissipation
simply from the time series of V . In this work we adopt two methods. In description (A), we treatthe time series of
1
V as that from a virtual single-RC circuit, and compute the current and therefore dissipation directly following the
1
equation of this simplified circuit. And in description (B), an effective dissipation can be derived using the ratio of
forward and backward transition probabilities in V over infinitesimal timesteps.
1
We first derive the steady-state probability distribution in V :
1
V′2 C˜ V′2
P (V )= dV P (V ,V )∼exp − 1 =exp − 1 1 , (14)
ss 1 Z 2 ss 1 2 (cid:18) M1I1Γ1(cid:19) 2kBT !
C C
where Mˆ I ≡ Mˆ−1, and C˜ ≡ C + 2 c is the effective capacitance. Based on Eq. 14, we can develop a naive
1 1
C +C
2 c
interpretation(“description(A)”),wherethemaskedcircuitistreatedasasingle-RCcircuitwithcapacitanceC˜ and
1
unmodifiedR adI . ThusR andI areneglectedintentionally. This effective single-RCcircuitcangivethe correct
1 1 2 2
steady-statedistributioninV . Alternatively,onecanregardthe time seriesofV asthatfromasingle-RCcircuit,as
1 1
the effective resistance and capacitance can be derived from its power spectrum, which can be shown to be identical
with R and C˜ , respectively.
1 1
The total entropy change of this gedanken single-RC circuit, during an infinitesimal timestep, is
P (V (t+dt))
dS˜(A) =dtV ˜i /T −k ln ss 1 , (15)
1tot 1 1 B P (V (t))
ss 1
where the first term on the RHS represents a “virtual” dissipation, as ˜i ≡ I −C˜ V˙ is the virtual current going
1 1 1 1
through R in this single-RC circuit. For the case of finite-time difference we have
1
1 τ
∆S˜(A) = dt(V I −V C˜ V˙ ). (16)
1tot,τ T 1 1 1d 1 1
Z0
Again one finds ∆S˜(A) to be Gaussian, while h∆S˜(A) i = I2R τ/T is the average virtual dissipation from R .
1tot,τ 1tot,τ 1 1 1
Using time-correlation functions, it is straightforwardto derive its variance:
h(∆S˜(A) )2i−h∆S˜(A) i2 =
1tot,τ 1tot,τ
2k I2R R C2 M M
B 1 1 τ − 1 c + 12 21 ·[λ (M +λ )e−τ/λ1 −λ (M +λ )e−τ/λ2] . (17)
T C +C M2 (λ −λ ) 1 22 2 2 22 1
(cid:26) 2 c 22 1 2 (cid:27)
Therefore,FTfailswiththeadoptionofsuchdissipationfunction,evenatthesmall-τ limit. Nevertheless,fromEq.17
one finds that this deviationbecomes less prominentat large τ. Moreover,one canalso show that the deviation from
FT diminishes in the weak-coupling regime, as the deviation in variance from 2k h∆S˜(A) i is proportional to C2.
B 1tot,τ c
1 Consider a Gaussian random variable x of average x0 and width σx. The corresponding symmetry function is Sym(x) =
ln[P(x)/P(−x)] = 2x0x/σx2. Thus Gaussianity is a sufficient condition of the FT-like behavior. FT is valid if σx2 = 2x0 (in di-
mensionlessunits;incases wherexmeans entropy thenFT validates ifσx2 =2kBx0),andforthecasewheretheobserved slopeisnot
equalto1,FTcanbeeasilyrestoredwiththerescaledvariablex′=(2x0/σx2)x.
4
III. REDUCED DESCRIPTION (B): TRACE-OUT APPROACH
Beside the above reduced description, one can define the effective dissipation function starting from the forward
transitionprobabilityof V overinfinitesimal timesteps. It canbe derivedby tracing outthe degree offreedom inV :
1 2
P (V (t+dt)|V (t))= dV (t)dV (t+dt)P (V~(t+dt)|V~(t))P (V~)/P (V )
F 1 1 2 2 F ss ss 1
ZZ
dt (M~I)TΓˆM~IV′ 2
∼exp − V˙ + 1 1 1
 2(M~1I)TΓˆM~1I " 1 M1I1Γ1 # 
≡exp[−dt(M˜V˙ +V′)2/(2Γ˜ )] (18)
 1 1 1 
to the lowest nonvanishing order, where M~I is a two-dimensional vector of elements MI and MI . Note that this
1 11 12
transition probability can be compared to that of a single-RC circuit with the aforementioned effective capacitance
C C MI Γ
C˜ ≡ C + 2 c , M˜ = R˜ C˜ = 11 1 = (MI )−1/(1+α), R˜ = R /(1+α), and I˜ = I (1+α), where
1 1 C2+Cc 1 1 (M~I)TΓˆM~I 11 1 1 1 1
1 1
M M R C2
α≡ 12 21 = 1 c . And the renormalized noise amplitude parameter is
M2 R (C +C )2
22 2 2 c
Γ˜ =(M~I)TΓˆM~IM˜2 =Γ /(1+α), (19)
1 1 1 1
which is smaller than Γ . Since exp[−C˜ V′2/(2k T)] = exp(−M˜V′2/Γ˜ ), this effective single-RC circuit also gives
1 1 1 B 1 1
the correct steady-state probability distribution in V .
1
In this effective single-RC circuit, the reversed transition probability is derived simply by replacing V˙ with −V˙
1 1
′
and V with V +V in Eq. 18. The net dissipation and total entropy change are
1 1 1d
P (V (t+dt)|V (t))
dS˜(B) =k ln F 1 1 =2k dt[V (V −M˜V˙ )/Γ˜ ] and (20)
1Q B P (V (t)|V (t+dt)) B 1 1d 1 1
R 1 1
P (V (t+dt))
dS˜(B) =dS˜(B)−k ln ss 1 =2k dtV (V −M˜V˙ )/Γ˜ , (21)
1tot 1Q B P (V (t)) B 1d 1 1 1
ss 1
respectively. Since dS˜(B) is linear in V and V˙ , the total entropy change is Gaussian. One can prove that FT is
1tot 1 1
satisfied for dS˜(B). However,violation still occurs in finite-time processes, where
1tot
h∆S˜(B) i=2k τV2/Γ˜ =I2R τ(1+α)/T , (22)
1tot,τ B 1d 1 1 1
and
2k I2R
h(∆S˜(B) )2i−h∆S˜(B) i2 = B 1 1 (1+α)2(τ −M )+R C˜
1tot,τ 1tot,τ T 11 1 1
(cid:26)
(λ −M )λ e−τ/λ1 −(λ −M )λ e−τ/λ2
+(1+α)2 2 11 1 1 11 2
λ −λ
2 1
(R C˜ )2 λ −M λ −M
− 1 1 2 11 e−τ/λ1 − 1 11 e−τ/λ2 . (23)
λ −λ λ λ
2 1 (cid:20)(cid:18) 1 (cid:19) (cid:18) 2 (cid:19) (cid:21)(cid:27)
Note that, on average,the reduced description (B) gives a larger total entropy change than description (A).
[1] G. E. Crooks, Phys.Rev. E60, 2721 (1999).
[2] U. Seifert, Rep. Prog. Phys.75, 126001 (2012).