Table Of ContentAn illustrative example of the relationship between dissipation and relative entropy
Jordan Horowitz1 and Christopher Jarzynski2
1Department of Physics, University of Maryland, College Park, MD 20742 USA
2Department of Chemistry and Biochemistry, and Institute for Physical Science and Technology,
University of Maryland, College Park, MD 20742 USA
(Dated: January 5, 2009)
Kawai, Parrondo, and Van den Broeck [Phys. Rev. Lett. 98, 080602 (2007)] have recently estab-
lishedaquantitativerelationship between dissipatedwork andamicroscopic, information-theoretic
9
measure of irreversibility. We illustrate this result using theexactly solvable system of a Brownian
0
0 particle in a dragged harmonic trap.
2
PACSnumbers: 05.70.Ln,05.40.-a
n
a
J
I. INTRODUCTION in Section II, we introduce our model at the beginning
5 of Section III. We then solve the model and illustrate
Duringathermodynamicprocessinwhichasystem,in Eq. 1 when the system evolves under Hamilton’s equa-
]
h contactwithathermalreservoir,evolvesfromonestateof tions (Section IIIA), and both overdamped and iner-
c thermal equilibrium A, to another B, the average work tialLangevindynamics (Sections IIIB and IIIC, respec-
e performed on the system must exceed the free energy tively). In Section IV we extend our analysis, and show
m
difference: W ∆F = F F [1]. The work in that the bound in Eq. 1 can be improved by specifying
B A
t- excess of ∆Fh , ii.e≥. the dissipated−work Wdiss = W the phase space density at two times, rather than one.
a ∆F,quantifiesthe amountofenergyirrhetrievaiblyhlostit−o Gomez-Marinet al. [15, 16] have alsostudied this model
st the surrounding thermal environment. Recently, Kawai, in the context of Eq. 1 and in Section IV we briefly dis-
t. Parrondo,andVandenBroeck[2]haverelated Wdiss to cuss the relationship between our results and theirs.
a h i
another measure of irreversibility, roughly speaking the
m
distinctionbetweenthe forwardandreversedirectionsof
-
d the “arrow of time”. Specifically, they have shown that II. BACKGROUND
n
o β Wdiss D(ρF ρR), (1)
h i≥ || ConsideraclassicalsystemwithN degreesoffreedom,
c
[ wsithieersedρeFscarinbdinρgRthaereetvimolue-tdioenpenofdetnhtepshyasstee-mspafcreomdenA- djuegsacrtiebemdobmyenthtae pcoo=rdinpa1,te.s..x,p=N{.x1L,.e.t.,zx=N}(xan,pd)codne--
1 { }
to B, and from B to A, respectively; and D(ρ ρ′) = note a point in phase space, and let H(z,λ) denote a
v
6 ρln(ρ/ρ′)denotesthe relative entropy[3],amea|s|ureof parameter-dependent Hamiltonian for this system. We
7 tRhe distinguishability between two distributions. When assume that this Hamiltonian is time-reversal invariant:
5 the system in question evolves deterministically, under H(z∗,λ)=H(z,λ),wheretheasteriskdenotestherever-
0 Hamilton’sequations,thenEq.1isanequality. However, sal of momenta, p p.
1. if the system description is coarse-grained, or explicitly The term therm→ody−namic process will indicate a se-
0 stochastic,thentherelativeentropyonlyprovidesalower quence of events whereby the system evolves in phase
9 bound on the dissipated work. space as the external parameter λ, is varied according
0 Connections between dissipation and temporal asym- to an arbitrary schedule, or protocol λt (also labeled as
:
v metrysimilartoEq.1havealsobeenestablishedbyother λ(t)), from an initial time t = 0 to a final time t = τ.
i authors. Maes [4] and later Maes and Netoˇcny´ [5] have Duringthis intervaloftime, the microscopicevolutionof
X
obtainedacorrespondencebetweenthermodynamicmea- the system is specified by a trajectory zt, or z(t), and
ar sures of dissipation and relative entropies between for- ρ(z,t) will denote the time-dependent phase space den-
ward and reverse distributions in path space; and Gas- sity describing an ensemble of such trajectories. As in
pard [6] has connected this result to the dynamical ran- Ref. [2] we will explicitly consider two such processes:
domnessthatcharacterizesnonequilibriumsteadystates. one defined by a forward protocol λF, during which the
t
In the context described in the previous paragraph, one parameter is varied from an initial value λF =A to a fi-
0
of us [7] has derived a relation analogous to Eq. 1, but nalvalueλF =B,theotherdefinedbyareverseprotocol
τ
expressed in terms of distributions in path space (see λR =λF , from λR =B to λR =A.
t τ−t 0 τ
Eq. 4). Prior to the start of either the forward or the reverse
ThegoalofthepresentpaperistoillustrateEq.1using process (t < 0), the system is brought to equilibrium by
the simple model of a particle inside a one-dimensional, weakcontactwitha thermalreservoiratinversetemper-
moving harmonic well. This model is both analytically ature β. As a result, the initial phase space density is a
tractableandexperimentallyrelevant[8,9,10,11,12,13, canonicaldistribution at λ=A or B. The system might
14]. After briefly reviewing the central result of Ref. [2] subsequentlybe thermallyisolatedfromthe reservoir,or
2
else itmightremainincontactwith the reservoir. Inthe the closer the value of D is to its upper bound β W .
diss
h i
former case, Hamilton’s equations govern its evolution We investigate this in Section IV, where we consider a
from t = 0 to t = τ, while in the latter case we will use generalization of Eq. 1 in which the microstate of the
stochastic dynamics to model the random effects of the system is specified at two instants in time, rather than
environment. In either situation the work performed on one. References [2, 15, 16, 17] contain a related analysis
the system during this interval of time is given by the specifically addressing the loss of microscopic informa-
following functional of the trajectory: tion during coarse-graining and how this loss affects the
τ ∂H value of the relative entropy.
W [z ]= dtλ˙ (z ,λ ). (2)
t t t
Z ∂λ
0
ForthecaseofHamiltoniandynamics,theaveragedis-
III. PARTICLE IN A MOVING HARMONIC
sipated work can be expressed as (see Ref. [2] for the
WELL
details of the derivation)
ρF(z,t) In this section we illustrate Eq. 1 through an explicit
β W = dzρ (z,t) ln
h dissi Z F (cid:20)ρR(z∗,τ t)(cid:21) calculation of the average dissipated work and relative
−
= D(ρ (z,t) ρ (z∗,τ t)). (3) entropy for a particle in a one-dimensional, moving har-
F R
|| − monic well. We begin by specifying the model and ob-
Sincerelativeentropyisameasureofthedistinguishabil-
taining useful preliminary results. Then in section IIIA
ity of two probability distributions [3], Eq. 3 relates the
we evaluate the averagedissipated work and relative en-
dissipationofenergytotheeasewithwhichaprocesscan
tropy for a system following Hamiltonian dynamics. In
be distinguished from its time-reversal[2, 15, 16, 17].
sections IIIB and IIIC we repeat the calculation with
Eq. 3 is an equality because the underlying Hamilto-
the system modeled by Langevin dynamics, in different
niandynamicsaredeterministic. Ifthesystemremainsin
limiting regimes.
contactwith the reservoirduringthe process,we instead
Our system is a particle of mass m trapped in a har-
obtainaninequality,Eq.1. Thereare twowaysto argue
monic well with spring constant k:
this, both of which make use of a key property of the
relative entropy between two distributions, namely that
p2 k
it decreases when the distributions are projected onto a H(x,p,λ)= + (x λ)2. (5)
2m 2 −
smaller set of variables [3]. When the system remains in
contact with a reservoir, then we can view the system
We will consider processes during which the center of
and the reservoir as two sub-systems that together form
the well is moved either rightward or leftward at con-
a large, isolated “super-system” to which Eq. 3 applies:
stant speed u. These correspond to forward and reverse
the dissipated work gives the relative entropy between
protocols, λ (t) = ut and λ (t) = u(τ t). Explicit
the two distributions in the full phase space. Upon pro- F R −
expressions for the initial equilibrium densities are given
jecting out the reservoir variables, the relative entropy
by the following Gaussians:
decreases, and we obtain Eq. 1, as discussed in Ref. [2].
Alternatively,wenotethatanequalityanalogoustoEq.3
can be formulated for distributions in path space, rather β k p2 kx2
ρ (z,0)= exp β + , (6a)
than phase space [7]: F 2πrm (cid:20)− (cid:18)2m 2 (cid:19)(cid:21)
β W =D( (γF) (γR)). (4) β k p2 k
h dissi PF ||PR ρR(z,0)= 2πrmexp(cid:20)−β(cid:18)2m + 2(x−uτ)2(cid:19)(cid:21).
HerethetrajectoryγF describestheevolutionofthesys-
(6b)
temduringagivenrealizationoftheforwardprocess,and
(γF) is the probability distribution of such trajecto-
F
rPies; γR and are defined in a similar manner for the For this system ∆F = 0 by translational symmetry,
R
P therefore for the forwardprocess
reverse process. Equation 4 holds for both deterministic
and stochastic evolution. If we now project from path
τ
tshpeacreeloantitvoepehnatrsoepsypadceec,reea.sge.s,PaFn(dγFw)e→agaρiFn(ozb,tt)a,inthtehne hWdissi=−ukZ0 dt[x¯F(t)−ut], (7)
inequality,Eq1. (References[15,16]refertothis projec-
tion procedure as “coarse-graining in time”.) In either where x¯F(t) is the average position during the forward
case, the decrease of relative entropy has a simple in- process.
terpretation: whenwediscardmicroscopicinformation– Since the time-dependent densities ρ (z,t) and
F
either about the state of the reservoir, or about states ρ (z,t) will prove to be Gaussians for all the cases con-
R
of the system itself at times other than the specified in- sidered in this paper (see Sections IIIA - IIIC), it is
stant, t – then we diminish our ability to distinguish be- useful here to establishuniform notationfor the descrip-
tween the forward and the reverse process. This sug- tion of two-dimensional Gaussian distributions f (z) =
G
gests that the more microscopic information we retain, f (x,p). Such a distribution is uniquely determined by
G
3
the moments (means and covariances), and the initial variances
m 1
σ2(0)= , σ2(0)= , σ2 (0)=0, (13b)
x¯ = dzfG(z)x, (8a) x β p βk xp
Z
whicharethesamefortheforwardandreverseprocesses.
p¯ = dzf (z)p, (8b)
G
Z Solving Eqs. 12 leads to a set of linear equations for the
positions and momenta in terms of their initial condi-
σ2 = dzf (z)(x x¯)2, (8c)
x Z G − tions. CombiningthesesolutionswithEqs.8and13leads
to the solutions
σ2 = dzf (z)(p p¯)2, (8d)
p Z G − sin(ωt)
x¯ (t)=u t , p¯ (t)=mu 1 cos(ωt) ,
F F
(cid:20) − ω (cid:21) −
σxp = dzfG(z)(x x¯)(p p¯). (8e) (cid:2) (cid:3)
Z − −
(14a)
An explicit expression for f (z) in terms of these mo-
G
sin(ωt)
ments is
x¯ (t)=u τ t+ , p¯ (t)=mu cos(ωt) 1 ,
R R
(cid:20) − ω (cid:21) −
1 1 (cid:2) (cid:3)
f (z)= exp (z z¯)T σ−1 (z z¯) , (9) (14b)
G
2π√detσ (cid:20)2 − · · − (cid:21)
wherez isavectorinphasespaceandσ isthecovariance m 1
σ2(t)= , σ2(t)= , σ (t)=0, (14c)
matrix: x β p βk xp
z = x , σ = σx2 σxp . (10) where ω2 =k/m.
(cid:18) p (cid:19) (cid:18)σxp σp2 (cid:19) We can now explicitly verify Eq. 1. The relative en-
tropy between the forward phase space density ρ (z,t)
Lastly,the relativeentropybetweentwoGaussiandistri- F
and the reverse phase space density ρ (z∗,τ t) is de-
butions, fG(z) and gG(z∗), is (see Eq. 3) termined by plugging Eqs. 14 into Eq.R11, −
1 detσ
D(fG(z)||gG(z∗))=−1+ 2(cid:20)ln(cid:18)detσg(cid:19)+Tr σg−1·σf∗ (cid:21) D(ρF(z,t)||ρR(z∗,τ −t))=βmu2 1−cos(ωτ) . (15)
f (cid:0) (cid:1) (cid:0) (cid:1)
1 Comparing this with
+ (z¯∗ z¯ )T σ−1 (z¯∗ z¯ ),
2 f − g · g · f − g
(11) β Wdiss =βmu2 1 cos(ωτ) , (16)
h i −
(cid:0) (cid:1)
where σ∗ = σ andall other elements of σ∗ are unal- computed from Eqs. 14a and 7, we find the predicted
tered [18x]p. − xp result β Wdiss =D(ρF ρR).
h i ||
A. Hamiltonian Dynamics B. Overdamped Langevin Dynamics
We now consider the case in which the system is ther- We now imagine that the system remains in contact
mally isolated from the reservoir after the initial equi- with the thermal reservoir throughout the process, and
libration stage, and thereafter evolves under Hamilton’s we will use Langevin dynamics to model the presence of
equations as the harmonic well is translated leftward or the reservoir. As discussed in Section II, Eq. 1 should
rightward, apply as a strict inequality in this case.
In this section we consider the overdamped limit, in
x˙ = p /m, p˙ = k(x ut), (12a) which the momentum effectively equilibrates instanta-
F F F F
− −
neously, and as a result the momentum does not con-
x˙ = p /m, p˙ = k(x uτ +ut). (12b)
R R R R
− − tributetotherelativeentropy. Wethereforefocusonthe
To solve for the evolution of the phase space densities position dynamics. For the forward process the Fokker-
ρF and ρR, we observe that since the initial density is Planck equation for ρF(x,t) is
Gaussian (Eqs. 6), and the equations of motion are lin-
∂ k ∂ 1 ∂2
ear (Eqs. 12), the distribution remains Gaussian for all
ρ (x,t)= [(x ut)ρ (x,t)]+ ρ (x,t),
times [19]. Thus,we need only determine the means and ∂t F γ∂x − F γβ∂x2 F
(co)variances as functions of time. From Eqs. 6 we have (17)
the initial means where γ is the friction coefficient.
TosolveEq.17,werecognizethataninitiallyGaussian
x¯ (0)=0, p¯ (0)=0, x¯ (0)=uτ, p¯ (0)=0, distribution will remain Gaussian for all time under the
F F R R
(13a) evolution of Eq. 17, as can be checked by substitution
4
(seeforexample[8,20]). Thus,ρ (x,t)isGaussianwith
F
mean and variance,
γu 1
x¯F(t)=ut− k (cid:16)1−e−kt/γ(cid:17), σF2 = βk. (18) 0.40
k l
For the reverseprocess we replacet with τ−t in Eq.17. 0.35 bWdiss
The solution is a Gaussian distribution with mean and
variance,
0.30
x¯ (t)=u(τ t)+ γu 1 e−k(τ−t)/γ , σ2 = 1 . D(rF||rR)
R − k (cid:16) − (cid:17) R βk
0.25
(19)
We now use these results to calculate the relative en-
tropy and average dissipated work. Using Eq. 11 we ob- 0.20
tain
2βγ2u2 0.15
D(ρF(x,t) ρR(x,τ t))= 1 0.0 0.5 1.0
|| − k (cid:20)
Time
k τ 2
e−kτ/2γcosh t .
− (cid:18)γ (cid:16)2 − (cid:17)(cid:19)(cid:21)
(20) FIG.1: Comparison ofβhWdissi(dashedline)andD(ρF||ρR)
(solid line) in the overdamped limit. All parameters (k, γ, u
and β) havebeen set to one in their respective units.
The average dissipated work is obtained by substituting
Eq. 18 into Eq. 7 and evaluating the integral:
β W =βγu2 τ γ 1 e−kτ/γ . (21) where Pλss(z) is the unique stationary state with fixed
diss
h i h − k (cid:16) − (cid:17)i external parameter λ and δP(z,t) is the first order cor-
rection to the phase space density [22]. Combining Eq.
Verifying Eq. 1 requires demonstrating that the aver-
23 with the definitions of average dissipated work (cf.
agedissipatedworkisalwaysgreaterthantherelativeen-
Eq. 2) and relative entropy, and taking the limit ǫ 0,
tropy for any values of system parameters. To begin, we leads to β W ǫ and D(ρ ρ ) ǫ2. In the a→bove
note that βhWdissi does not depend on t and D(ρF||ρR) model, λ(th)=diusstiw∼ith ǫ=u. F|| R ∼
obtains its maximum value D at time t=τ/2. Com-
max
bining Eqs. 20 and 21, we get
β W D(ρ ρ ) β W D C. Full Phase Space Langevin Dynamics
diss F R diss max
h i− || ≥ h i−
βγ2u2
= ζ 3 e−ζ/2 1 e−ζ/2 In this section we analyze the same stochastic system
k h −(cid:16) − (cid:17)(cid:16) − (cid:17)wiithoutassumingthe overdampedlimit. (We do assume
0,
≥ thatthemotionisnotcriticallydamped,i.e. γ2 =4mk.)
(22) 6
Using the same notation as the previous section (IIIB)
the Fokker-Planck equation for the phase space density
where ζ = kτ/γ is the scaled time. Using introductory
of the forward process is
calculustechniques,itiseasytoverifythatthebracketed
quantity in Eq. 22 is non-negative for any ζ 0. As an
∂ p ∂
≥
example, we plot βhWdissi and D(ρF||ρR) in Fig. 1 as ∂tρF(z,t)=− m∂xρF(z,t)
functions of time for k, γ, u and β all set to one.
∂ γp
In the quasi-static limit, namely u 0 with uτ fixed, + k(x ut)+ ρF(z,t) (24)
both the dissipation and relative entr→opy approachzero, ∂ph(cid:16) − m(cid:17) i
althoughatdifferentrates. Specifically,fromEqs.20and γ ∂2
+ ρ (z,t).
21, β W u and D(ρ ρ ) u2. This limiting be- β ∂p2 F
diss F R
h i∼ || ∼
havior can be justified on general grounds since we are
considering continuous Markovian stochastic processes While Eqs. 24 can be solved exactly, the solution is
[21]. A heuristic argument is as follows: consider such complicated and unilluminating. An analytic solution is
a continuous Markovianstochastic process perturbed by presented in Appendix A; here, we illustrate Eq. 1 by
varying anexternalparameter λ accordingto a specified plotting the ratio β Wdiss /D(ρF ρR) in Fig. 2 for var-
h i ||
protocol λ(t). In the quasi-static limit, i.e. λ˙ ǫ 1, ious values of the friction coefficient γ. Roughly speak-
the phase space density is approximately ∼ ≪ ing, γ measures the coupling between the system and
the reservoir: for larger γ, the reservoir and system in-
P(z,t) Pss (z)+ǫδP(z,t), (23) teractmorestrongly;whereas,whenγ 0,thereservoir
∼ λ(t) →
5
Section IIIB. The goal is to determine the pairwise
probability P(x ,t ;x ,t ). For the forward process,
1 1 0 0
we can decompose the joint probability distribution
1.00
into a conditional probability to be at x at t given
1 1
the system was at x at t , ρ (x ,t x ,t ), and a
0.99 g=0.1 0 0 F 1 1| 0 0
one-time probability ρ (x ,t ), i.e. P (x ,t ;x ,t ) =
F 0 0 F 1 1 0 0
0.98 ρF(x1,t1 x0,t0)ρF(x0,t0). The expression for ρF(x0,t0)
|
was determined in Section IIIB (see Eq. 18). The con-
0.97 ditional probability is computed by solving the Fokker-
Planck equation (see Eq. 17) with the initial condition
0.96 g=5
δ(x x ) at t , i.e. the initial density is Gaussian with
0 0
x¯(t −) = x and σ2(t ) = 0. At time t , the mean and
0.95 0 0 x 0 1
variance for the forward process conditional probability
are
0.94
0.93 x¯(t1)=e−k(t1−t0)/γx0+u t1 t0e−k(t1−t0)/γ
0.0 0.5 1.0 (cid:16) − (cid:17) (25a)
γu
Time 1 e−k(t1−t0)/γ ,
− k (cid:16) − (cid:17)
FIG.2: ComparisonofβhWdissi/D(ρF||ρR)forvariousvalues
1
ofthefrictioncoefficientγ;thesolidlinesare,frombottomto σ2(t )= 1 e−2k(t1−t0)/γ . (25b)
top, γ =5 and γ =0.1. The dashed line represents Hamilto- x 1 βk (cid:16) − (cid:17)
nian dynamics(γ =0) with βhWdissi=D(ρF||ρR). Allother
The pairwisedistributionisthe productofGaussiandis-
system parameters (m, k, u and β) have been set to one in
their respective units. tributions; as such, it is Gaussian as well. A simple, yet
lengthy calculation shows that the means and variances
of the pairwise distribution for the forward process are
andthesystemdecouple,withtheresultthatthesystem
etivcoHlvaemsiinltdoenpiaennddeynntlaymoifcst.hFeirgeusererv2oicrl,eaurnldyesrhdowetsertmhaintiass- x¯F =(cid:18)xx¯¯10 (cid:19)=(cid:18)uutt10−− γγkkuu(cid:0)11−−ee−−kktt01//γγ(cid:1)(cid:19) (26a)
γ decreases, the relative entropy approaches the average (cid:0) (cid:1)
dissipated work. Decreasing γ weakens the interaction
bmeitcwroesecnoptihceinsyfostremmataionndltehaekinregseirnvtooirt,hreesrueslteirnvgoiirnvlaersis- σF =(cid:18)σσxx20x01 σσxx02x11 (cid:19)= β1k (cid:18)e−k(t11−t0)/γ e−k(t11−t0)/γ (cid:19)
ables. The result is more informationis contained in the (26b)
system variables, reflected by an increase in the relative Repeating the abovecalculationforthe reversejointdis-
entropy. The flow of microscopic information between tribution leads to similar results.
system and reservoir degrees of freedom and its relation The next step is to determine the relative entropy.
to dissipation has previously been discussed in the con- We are interested in comparing the pairwise probabil-
text of other models [15, 17]. ity for the forward process PF(x1,t1;x0,t0), with the
time-reversedpairwiseprobabilityforthereverseprocess
P (x ,τ t ;x ,τ t ). If we allow t and t to be
R 0 0 1 1 0 1
− −
arbitrary,then from Eq. 11 the relative entropy is
IV. TWO-TIME PHASE SPACE DENSITIES
2βγ2u2/k
Intheprevioussections,thephasespacedensitieswere D(PF||PR)=1 e−2k|t1−t0|/γ (27)
evaluated at one particular time. For Hamiltonian dy- −
namicsthis isenoughinformationtodetermine the aver- ×(cid:16)α20+α21−2e−k|t1−t0|/γα0α1(cid:17)
agedissipated work. Incontrast,whenthe dynamics are
stochasticthe phasespace densities alonedo notcontain where
enough information to determine the average dissipated
k τ
work. In this section we address this issue by investigat- α =1 e−kτ/2γcosh t , i=0,1. (28)
i i
ingtheimplicationsofspecifyingthephasespacedensity − (cid:18)γ (cid:16)2 − (cid:17)(cid:19)
attwotimes. LetP(z ,t ;z t )bethe“two-time”prob-
1 1 0, 0
ability for the system to be at z0 at time t0 and z1 at a Figure 3 compares the “two-time” relative entropy
later time t1. We will explicitly demonstrate that two- D(PF PR), “one-time” relative entropy D(ρF ρR), and
|| ||
timedistributionsprovideabetterboundfortheaverage the averagedissipated work. For D(P P ), we fix t =
F R 0
||
dissipated work: βhWdissi≥D(PF||PR)≥D(ρF||ρR). τ/4. WeseethatβhWdissi≥D(PF||PR)≥D(ρF||ρR)for
For clarity we consider the overdamped limit as in alltimest . Thus,byspecifyinganadditionaltimeinour
1
6
is not true for stochastic systems. Calculating the dis-
sipation from microscopic information in stochastic sys-
temsrequiresknowledgeoftheentireevolutionofthesys-
tem, both for the forward and reverse processes. Partial
0.40
knowledge gives only a lower bound on the dissipation.
k l We have seen that the tightness of the lower bound is
0.35 bWdiss correlated with the amount of information known about
the system’s evolution. Specifically, when the system is
D(PF||PR)
loosely coupled to a reservoir, very little information is
0.30
lost to the reservoir. As a result the relative entropy
between the system’s forward and reverse phase space
0.25 densities reasonablyapproximatesthe dissipationin this
limit. The lower bound can also be tightened using a
D(rF||rR)
multi-time relative entropy where the microscopic state
0.20 of the system is specified many times along the system’s
trajectory.
0.15
0.00 0.25 0.50 0.75 1.00
Acknowledgments
t1
We thank S. Vaikuntanathan and A. Ballard for a
FIG. 3: Comparison of D(PF||PR) (dot-dashed line), critical reading of this manuscript and the University of
D(ρF||ρR)(solidline)andβhWdissi(dashedline)intheover- Maryland for their generous support.
dampedlimit. Allsystemparametershavebeensettoonein
their respective units.
APPENDIX A: PHASE SPACE DENSITY FOR
SECTION IIIC
probability distributions we have improvedthe bound in
Eq. 1.
In this appendix, we solve Eq. 24 for the full phase
This resultis aninstance ofthe coarse-grainingproce-
space density of Section IIIC. To begin, the coefficients
dure outlined in Section II (see the discussion below Eq.
4),wherethepathspaceprobabilitydistribution (γF) of Eq. 24 are linear in x and p. Consequently, as in the
F
P previous sections, an initial Gaussian distribution will
has been projected down onto a two-time phase space
remain Gaussian. Again we must determine the means,
density. Projecting onto a two-time phase space density
x¯ and p¯, and covariance matrix σ as functions of time.
eliminates less microscopic information than projecting
Combining Eq. 24 with the derivatives of Eqs. 8 we find
ontoaone-timephasespacedensity;hence,thetwo-time
the equations of motion for the means and variances of
relative entropy is greater than the one-time relative en-
the forward process
tropy.
In a related independent analysis, Gomez-Marinet al. dx¯ p¯ dp¯ γ
F F F
= , = k(x¯ ut) p¯ , (A1a)
[15, 16] used the presentmodel to investigatethe behav- dt m dt − F − − m F
ior of the “N-time” relative entropy D . By evaluating
N
awDseNrNeatabNleetq.ouWadleelymnsoopntaesctterhadattetimothuearst,tiwβ.eoh.W-ttiid+mis1es−ir−etilaD=tiNvτe/∼Nen,t1rt/ohNpeyy2 ddσtx2 = m2 σxp, ddσtp2 =−2mγσp2−2kσxp+ 2βγ, (A1b)
agrees→wi∞th theirs: choosing t1 =τ and t0 =0 in Eq.27, dσdtxp =−mγ σxp−kσx2+ m1 σp2.
we recover Eq. 30 of Ref. [15] and Eq. 22 (with n = 1)
of Ref. [16]. The reverse process phase space density is described by
a similar set of equations.
The solutions of Eqs. A1a and their reverse process
V. CONCLUSION counterparts are
x¯ (t) = Aer+t+Ber−t+u(t γ/k), (A2a)
F
The average energy dissipated by a thermodynamic −
p¯ (t) = mr Aer+t+mr Ber−t+mu, (A2b)
process is related to how distinguishable the process is F + −
from its time-reversal, i.e. how well one can discern the x¯R(t) = Aer+t Ber−t+u(τ t+γ/k),(A2c)
− − −
“arrowoftime”. Thispaperprovidesapedagogicalillus- p¯ (t) = mr Aer+t mr Ber−t mu, (A2d)
R + −
tration of this idea using an exactly solvable model. For − − −
where
deterministic Hamiltonian systems, complete knowledge
of the forward and reverse phase space density at any γ γ2 4mk
one time is enough to determine the dissipation. This r± = − ±p2m− (A3)
7
and Using Eq. 7, the average dissipated work is
A mu γr /k+1
= − + (A4)
(cid:18)B (cid:19) γ2 4mk (cid:18) γr−/k 1(cid:19)
− − −
p
are determined by the initial conditions (Eq. 13a). The
solutions to Eq.A1b with initial conditions givenby Eq. A B
W =γu2τ +uk 1 er+t + 1 er−t .
13b are h dissi (cid:20)r − r − (cid:21)
+ (cid:0) (cid:1) − (cid:0) (cid:1)
(A6)
m 1
σ2(t)= , σ2(t)= , σ (t)=0. (A5) From Eq. 11, the relative entropy is
x β p βk xp
1 2
D(ρ ρ )= βk A er+t+er+(τ−t) +B er−t+er−(τ−t)
F R
|| 2(cid:26) h (cid:16) (cid:17) (cid:16) (cid:17)i
(A7)
2
+βm r A er+t er+(τ−t) +r B er−t er−(τ−t) .
+ −
h (cid:16) − (cid:17) (cid:16) − (cid:17)i (cid:27)
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