Table Of ContentDissipation dynamics with two distinct chaotic baths
Diptanil Roy1 and A.V. Anil Kumar1
1School of Physical Sciences, National Institute of Science Education and Research, HBNI, Jatni - 752050, India
(Dated: January 30, 2017)
Dissipation using a finite environment coupled to a single harmonic oscillator have been studied
quite extensively. We extend the study by looking at the dynamics of the dissipation when we
introduce a second bath of N identical quartic systems different from the 1st bath. We look at the
energy flow into the environment as a function of the chaotic parameters of the bath and also try
to develop a linear response theory to describe the system. The energy flow is always more to the
more chaotic system irrespective of the initial energy of the baths.
PACSnumbers: 05.20.Gg,05.45.Ac,05.40.Jc
7
1
I. INTRODUCTION II. THE MODEL
0
2
Ourmodelcomprisesofaparticle,withgeneralisedco-
n
ordinates (q, p) in a one dimensional harmonic oscillator
a
J potentialwithangularfrequencyω0. Theparticleiscon-
7 Thedissipationofenergyoccursindiversesystemsand nected to two baths comprising of N quartic oscillators
2 itisoneofthemostfundamentalprocessesindissipative each. Thetwobathsarenotconnectedtoeachother,but
dynamicalsystems. IntheLangevindescriptionofBrow- are coupled indirectly through the harmonic oscillator.
] nian motion, the damping force together with the ran- The quartic oscillators are two dimensional and the n
D th
domlyfluctatingforceareusedtomodeldissipation[1,2]. particleofthei bathischaracterisedbythegeneralised
th
C These forces relate the effect of the collisions of the sys- coordinates(q(cid:126) =x ˆi+y ˆj, p(cid:126) =p ˆi+p ˆj). The
. tem with the particles in the thermal bath and are re- interaction betnwieen tnhie batnhis anndi thexsnyistemyisnithrough
n
i alted by the dissipation-fluctuation theorem[2]. The dis- a coupling term of the form HI, where λN = √λ is a
l sipative dynamics of physical systems have been stud- N
n measure of the effective coupling [9]. The Hamiltonian
ied extensively[3–10]. In modelling energy dissipation,
[ govering the dynamics of the system can be written as
these studies used a small system couple to the envi-
1
ronment. The environment can be of two types : one
v H =H +H +H +H (1)
with an infinite collection of modes[3] and another with HO E1 E2 I
7
a small number of chaotic modes[4, 5, 9, 10]. These in-
9
0 vestigations have led to several interesting results. For
p2 mω2q2
8 example, Wilkinson[4] showed that the rate of exchange H = + 0
0 of energy between the system and the chaotic bath de- HO 2m 2
01. ptheendcshaoontitchebactlhas.siTcahlemraotteioonfoefntehrgeypaexrtcihcalensgceoimncprreiasisnegs HEi = (cid:88)N (cid:34)p2xni +2 p2yni + a4i(x4ni +yn4i)+ x2ni2yn2i(cid:35)
7 when the bath shows chaotic motion. Later Marchiori n=1
1 and de Aguiar showed that energy dissipation occurs in N
(cid:88)
v: the chaotic regime even if the bath is composed of small HI = λNq(xn1 +xn2)
i number of particles[9]. Xavier et al.[10] also improved n=1
X
upon the model in [9] and showed that the damping rate
r can be expressed in terms of the mean Lyapunov expo- The total Hamiltonian is conservative, but the system
a
nent of the chaotic bath. Also they have shown that the can exchange energy with the baths and hence behave
dissipationismoreeffectiveifthereisresonancebetween dissipatively. The non-linear dynamics of the baths is
the system and bath frequencies. determined completely by the parameters ai where the
systems are integrable for a = 1.0, mildly chaotic for
i
a = 0.1 and strongly chaotic for a = 0.01. We numeri-
i i
cally solve the 8N +2 equations of motion arising from
In this paper, we extend this study to the case of a the Hamiltonians H , H , H and H respectively
HO E1 E2 I
small system coupled to multiple baths. The purpose and compare our analytical results to them. The chaotic
of this investigation is to study the dissipative dynam- dynamics of the baths acts as a sink of energy for the
ics of the system if it is connected to two distinct baths system. A temperature for the environment can be de-
with different chaoticity. We use linear response theory fined by using the general equipartition theorem, where
following the approach presented in [9]. We also carry every quadratic degree of freedom has a contribution of
our numerical simulations and compare the theoretical k T/2 to the total energy E. It is clear here that the T
B
results with those obtained from simulations. represents the equilibrium temperature.
2
FIG. 1: The correlation functions C(i)(t,0) for the two sets of baths for different set of initial parameters. C(i) deviates from
N N
the delta function approximation in the mixed and integrable regimes.
Before we proceed to the results for our model, it is III. THEORY
imperative to discuss the behaviour of a particle in har-
monic oscillator potential coupled to only one bath. We
Wewilltrytounderstandtheflowofenergytothetwo
will compare and contrast our results to those provided
different baths using linear response theory (LRT) [11].
in [9, 10]. When a=1.0, the system is in the integrable
We will follow the formalism developed in [9] to write
regimeandtheenvironmenthasverylittleeffect,causing
the final equation of motion for the system. The HO in
a slight decrease in the oscillator energy. With a = 0.5,
hamiltonian 1 satisfies the following equation of motion
theenergylossismoreandthereareslightlypronounced
oscillations. The equilibrium energy value for such cases
N
is 0.8E0 [9]. For example, for a bath with N =100 QOs q¨+ω2q =−λN (cid:88){x(1)(t)+x(2)(t)}≡−λNX(t) (2)
[9, 10] and a = 1, the energy of the system decreases 0 m n n m
n=1
a bit from E (0) and performs tiny oscillations around
S
the decreased value. For a = 0.1, the energy decreases
WeconsidertheHOtobeperturbedbyanexternalforce
further with more prominent oscillations with increasing given by −λNX(t). Underthe assumptionthat thebath
time. For a = 0.01, the situation however is completely m
coordinatesarechaotic,wecanreplaceX(t)with(cid:104)X(t)(cid:105).
different. The energy decay for such a system is expo-
LRT has been used to determine (cid:104)X(t)(cid:105) in terms of the
nential and can be fit to E (t)≈E (0)e−γt where γ is a
S S response function. The dynamics of the system is deter-
function of the bath properties and ω .
0 mined by the Liouville equations. The integral form of
For small values of N, the indirect coupling does not the Liouville equation is given by
have much effect on the system unless the coupling is
stronglychaotic. Forexample,whenthecouplingparam- (cid:90) t
ρ(t)=ei(t−t0)L0ρ(t )+i ei(t−s)L0L (s)ρ(s)ds (3)
eter is set in the chaotic regime, we cannot characterise 0 I
the behaviour by a single realisation for 1 ≤ N ≤ 6 be- t0
causeoflargefluctuations. For7≤N ≤20however, the whereL andL aretheLiouvilleoperators. TheHamil-
0 I
onset of dissipation and the transition to an exponential tonian 1 can be estimated using
decayareapparent, thoughtheenergyfluctuationisstill
large. With increasing N, the system starts to behave
H =H (Q,P)+H (Q,P,t) (4)
differently, allowing dissipation to occur. The effect of E I
different values of the coupling are discussed above.
whereQandP representtheentiresetofcanonicalvari-
Asisapparentfromthenumericalresults, thevalueof ables of the environment and H (Q,P,t)=A(Q,P)χ(t)
I
ES(t) for large N becomes independent of N over large where A(Q,P) is the corresponding displacement (Aˆ =
times. This is possible only when the coupling term λN − ∂ Hˆ(q,p;x)). Undertheapproximationofasmallper-
fallswithN asN →∞. Foronebath,ithasbeenshown tu∂rbxation, Eq. 3 can be expanded to 1st order in L as
I
in [9] that λN = √λ using Linear response theory. For
N
large N, the system equilibrates with the environment (cid:90) t
and the equilibrium energy distribution is Boltzmann- ρ(t)=ρ(t0)+ ei(t−s)L0{HI(s),ρ(t0)}ds (5)
like. This allows us to define a temperature; however it t0
is not possible in the regular or mixed regimes.
3
FIG. 2: The numerical and theoretical curves for 4 different sets of initial parameters. (a) a = 0.01; a = 0.05; E = 1;
1 2 1
E = 5 (b) a = 0.01; a = 0.05; E = 5; E = 1 (c) a = 0.01; a = 0.1; E = 5; E = 1 (d) a = 0.01; a = 0.1; E = 1;
2 1 2 1 2 1 2 1 2 1 2 1
E =5
2
The ensemble average (cid:104)X(t)(cid:105) can thus be written as or
(cid:90) t 5 d t−s d2
(cid:104)X(t)(cid:105)=(cid:104)X(t)(cid:105) −λ φ (t−s)q(s)ds (6) φ(i)(t−s)= C(i)(t,s)+ C(i)(t,s) (10)
0 N XX xx 4ds N 4 dsdt N
0
where (cid:104)X(t)(cid:105)0 is equal to 0 due to the parity of HE. where C(i)(t,s) is the correlation function given by
Therefore, Eq. 2 reduces to N
q¨+ω2q ≈ λ2NN (cid:90) t(φ(1)+φ(2))q(s)ds (7) C(i)(t,s)=(cid:42)(cid:88)N xn(i)(t)xn(i)(s)(cid:43) (11)
0 m 0 xx xx N E(i)(0)
n=1 QS
Here φ (t−s) = (cid:104){X(t),X(s)}(cid:105) and from [9], it fol-
XX 0
lows that Following [9], we make this approximation
φXX(t−s)=N(φ(x1x)+φ(x2x)) (8) C(i)(t,s)≈µ(i)Nδ(t−s) (12)
N E¯i
Here,φ(i) istheresponsefunctionobtainedin[12]andis
xx
given by whereA(i) =µ(i)N isthemaximumamplitudeofthecor-
N E¯i
relation function. In the time scale of dissipation, this is
5
φ(i)(t−s)= (cid:104)x(i)(t)P(i)(s)(cid:105) avalidapproximation whenthequartic oscillatorsarein
xx 4E(i)(0) x 0 the chaotic regime. As we transition to the mixed and
QS
t−s integrable regimes, the approximation fits poorly, how-
+ 4E(i)(0)(cid:104)Px(i)(t)Px(i)(s)(cid:105)0 (9) everwewillconsiderthedeltafunctionapproximationin
QS the mixed regime too for ease of calculations.
4
From Eq. 7 and Eq. 12, we obtain
q¨+ω2q+γ q˙=0 (13)
0 T
where
7λ2 (µ(1)+µ(2))N
γ = N E¯1 E¯2 (14)
T 4m
Theγ weobtaintheoreticallycompareswellwiththe
T
numerical results (Fig. 2) we obtained.
IV. NUMERICAL SIMULATIONS
We have also simulated the time evolution of our sys-
tembysolving8N+2equationsofmotionobtainedfrom
the Hamiltonian in Eq. 1 numerically. The second or-
der differential equations were solved using fourth order
Runge-Kutta algorithm in a microcanonical ensemble.
The time step has been taken to be 0.4 in each case.
For all the simulations, the value of λ is taken as 0.01.
The mass m is taken as 1 for all the oscillators and the
angular frequency of the harmonic oscillator ω is taken
0
tobe0.3. Thetimeismeasuredintermsoftheharmonic
oscillator time period τ ≈20.93.
V. RESULTS AND DISCUSSION
The results for our model are obtained both from Lin-
ear response theory and numerical simulations. Figure
2 shows the energy of the harmonic oscillator for four FIG.3: Theenergyflowintotwodifferentsetsofbaths,both
differentsetofinitialconditionsobtainedfrombothsim- in chaotic regime . The central HO has an initial energy of
10.
ulations and linear response theory. The parameters for
the harmonic oscillators are same for all the four graphs.
The thoeretical and numerical results are in good agree-
the more chaotic system has less initial energy and less
ment with each other. The deviations can be attributed
chaotic system has more initial energy. As evident from
to the δ-fucntion approximation for the correlation func-
thefigure,moreofthedissipatedenergyfromthesystem
tion used in the liner response theory. In the case of one
goes to the more chaotic system. Figure 2(b) shows an
chaotic bath, this has been improved upon by Xavier
interesting behaviour in the energy flow. In figure 3, the
et al.[10] by incorporating the history and frequency de-
more chaotic system has more energy and less chaotic
pence of the system. We intend to incorporate these in
system has less energy. So natuarally one would expect
our model and the results will be reported elsewhere.
more energy to flow to the bath which has lesser energy(
Figure 2 confirms that the harmonic oscillator is dissi-
andlesschaotic). However,moreofthedissipatedenergy
pative and energy flow occuring from the system to the
has flown into the more chaotic system even though its
bath. Also we have done simulations in which both the
energy is higher than the other bath.
baths are not chaotic. In this case, the harmonic oscilla-
tordoesnotdissipateenergyconsiderablyintothebaths Figure 4 shows two graphs which depicts the enrgy
and the energy dissipation happens only if at least one flow into the two different baths, one bath in the chaotic
of the baths is in the chaotic regime. regime and the other in the mixed regime. Here also it
Next, we have analysed the energy flow from the har- is evident that irrespective of the initial energy of the
monic oscillator to the chaotic bath. For this, we have baths, the energy dissipated from the system flows more
calculated the total energy of the system and each bath into the chaotic system. This has been the case with all
at regular intervals. This has been shown in Figure 3 the simulations, we have done in our model for different
and Figure 4. In Figure 3, we have plotted the total parameters. Also, it has been observed that the bath in
energy of the system and the two baths, both of which the mixed regime attained the steady state much faster
are in chaotic regime, but with different chaotic param- than the chaotic bath.
eter and initial energy, versus time. In Figure 3(a) the This result is rather surprising and nonintuitive. Us-
5
temperature for the environment. Therefore, each bath
contributes E(i) = NkBT to the total energy. There-
2
fore,differentinitialenergiesforthetwobathsessentially
means coupling the central HO to two baths at two dif-
ferenttemperatures,thebathwithhigherenergybeingat
a higher temperature. In such case, normally one would
expect more energy to flow to the bath at lower temper-
arute. Thishasbeenthecaseifthelowtemperaturebath
is more chaotic. However, the reverse happens if the low
temperature bath is less chaotic.
The seemingly counterintuitive result may be ex-
plained in terms of the entropy of the system. The more
chaotic system will have more entropy and the energy
will flow into the more chaotic bath such that the to-
tal entropy of the system will increase. This also causes
the more chaotic system to take more time to reach the
steady state, which is evident in Figure 4. More anal-
ysis are being carried out to quantitatively justify this
observed result.
VI. CONCLUSIONS
The dissipative dynamics of a one dimentional har-
monic oscillator coupled to two distinct baths has been
investigated using linear response theory and numerical
simulations. It has been shown that the harmonic os-
cillator dissipates energy into the baths if the baths are
chaotic. The numerical and theoretical results compares
well. We have also analysed the dissipative energy flow
into the baths from the system. It has been observed
that more energy flows into the more chaotic system ir-
FIG.4: Theenergyflowintotwodifferentsetsofbaths,one
respective of the initial energy of the baths. This rather
in chaotic and one in mixed regime. The central HO has an
surprising result may be explained based on the maxi-
initial energy of 10.
mization of entropy of the system. One might be able to
justify this more precisely by calculating the dynamical
ing the general equipartition theorem, we can define a entropy of the system for different sets of parameters.
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