Formulation of statistical mechanics for chaotic systems

PRAMANA c Inian Acaemy of Sciences Vol. 72, No. 2 journal of February 29 physics pp. 315 323 Formulation of statistical mechanics for chaotic systems VISHNU M BANNUR 1, an RAMESH BABU THAYYULLATHIL 2 1 Department of Physics, University of Calicut, Kozhikoe 673 635, Inia 2 Department of Physics, Cochin University of Science an Technology, Kochi 682 22, Inia Corresponing author. E-mail: vmbannur@yahoo.co.in MS receive 14 July 28; revise 21 August 28; accepte 26 August 28 Abstract. We formulate the statistical mechanics of chaotic system with few egrees of freeom an investigate the quartic oscillator system using microcanonical an canonical ensembles. Results of statistical mechanics are numerically verifie by consiering the ynamical evolution of quartic oscillator system with two egrees of freeom. Keywors. Statistical mechanics; chaotic system; quartic oscillators. PACS Nos 5.45.-a; 5.2.Gg; 5.7.-a 1. Introuction The stuy of statistical mechanics an thermoynamics of chaotic systems with few egrees of freeom is very important in unerstaning its various formal aspects from a ynamical point of view [1] an for the stuy of chaotic system using the well-evelope concepts of statistical mechanics [2,3]. Since the trajectory of a chaotic system is almost ergoic in phase-space, it can be approximately escribe by the principles of statistical mechanics. The chaotic systems like quartic oscillator (QO), Henon Heiles oscillator (HHO) etc., with two egrees of freeom exhibit chaos epening on the value of parameters of the system. Hence it is possible to stuy the thermoynamics of the system from the ynamical point of view. Issues relate to the efinition of various thermoynamical quantities such as entropy, temperature etc., are iscusse in the literature [2 4]. There are also other highly ebate issues like, the stuy of Fourier heat law, thermalization of oscillator chain etc. [5 12] using chaotic systems. We know that in a thermoynamical system, as the number of microsystems N the system is riven to ergoicity an the statistical property comes from the implicit assumption of collisions. On the contrary, in a chaotic system, N is finite an the nonlinear interactions present in the Hamiltonian rive the system to an almost ergoic region, provie the system is in a chaotic region. There may be few non-chaotic islans epening on 315

Vishnu M Bannur an Ramesh Babu Thayyullathil the values of the parameters, where the system may not show any ergoicity an the statistical properties an thermoynamics of the system may not have much meaning in that region. An initial stuy of statistical mechanics of chaotic system was carrie out by Berichevsky an Alberti [2] using Henon Heiles oscillator an later it was extene to quartic oscillator [3] also. Various issues like concepts of temperature, entropy an istribution function an equipartition of energy were stuie in etail an also verifie numerically using the formulation of microcanonical ensemble. Here, we again strengthen the iea of statistical mechanics of chaotic systems using Kinchin s formulation base on microcanonical ensemble [13]. Further, we exten the stuy to canonical ensemble of such a system an as an example, we consier QO an obtain various thermoynamic quantities an the results are verifie numerically. 2. Henon Heiles oscillator an quartic oscillator First we briefly iscuss QO an HHO moels, stuie extensively in the context of chaos an statistical mechanics of chaos [2,3]. Both are Hamiltonian systems with two egrees of freeom an the Hamiltonians are given by an H = (p2 1 + p 2 2) 2 H = (p2 1 + p 2 2) 2 + (1 α) (q1 4 + q 4 12 2) + 1 2 q2 1q2, 2 (1) + q2 1 2 + q2 2 2 + q2 1q 2 1 3 q3 2, (2) for QO an HHO respectively. Here q s an p s are generalize coorinates an momenta respectively an α is a parameter. QO is chaotic except for the values of α =, 2 an, an is highly chaotic for α close to 1 as shown in refs [14 16]. HHO is chaotic for energy E = 1/6 an evelops non-chaotic islans as the energy is ecrease. In orer to stuy the statistical mechanics using microcanonical ensemble first we nee to evaluate the boune phase-space volume for a given energy E an it is given by Σ(E) = p 1 p 2 q 1 q 2 = CE 3/2, (3) H E for QO with α 1 an for HHO ( Σ(E) = πe 2 1 + E 2 + 35 ) 32 E2 +, (4) as iscusse in etail in ref. [3]. Note that for α = 1 the phase-space volume of QO is infinite an hence unefine. In our numerical work we chose α =.99. In ref. [3] we also iscusse the generalize QO with N egrees of freeom for the Hamiltonian 316 Pramana J. Phys., Vol. 72, No. 2, February 29

Statistical mechanics for chaotic systems H = 1 2 N p 2 i + 1 2 i=1 N α ij qi 2 qj 2, (5) i,j=1 where α ij are parameters. This system reuces to our earlier N = 2 quartic oscillator if α 11 = α 22 = (1 α)/6 an α 12 = α 21 = 1/2. Phase-space volume for this system is given by Σ(E) = p 1...p N q 1...q N = C 1 E 3N/4, (6) H E where C 1 is a constant. 3. Microcanonical ensemble In the microcanonical ensemble, one consier a system with constant energy an in our case we consier a Hamiltonian system like HHO or QO with constant energy. Depening on the value of the parameters, the system may exhibit chaos an their phase-space trajectory is almost ergoic in the whole or some region of the phasespace. In those chaotic regions we may have averages of any observable O given as O t = O ph = O en, (7) which follows from ergoic theorem. Here O t, O ph an O en refers to time, phase-space an ensemble averages respectively. For example, the average of momentum square of ith egrees of freeom p 2 i approaches the same value on both time average an phase-space average, as verifie in ref. [3]. The phase-space average is efine as O ph 1 Σ O, (8) Γ E H E+ where Σ, Γ an Ω ( Σ/) are the number of microstates in phase-space hypersphere, hypershell an on the hypersphere surface respectively. Using the above efinition of average we now have the equipartition theorem, [ ] 1 ln Σ p 2 1 = p 2 2 = = T B. (9) Here T B may be ientifie as a temperature by consiering a thought experiment of equilibration of two systems in thermal contact. Suppose we have two systems 1 an 2 an they are brought into thermal contact. After equilibration, the average momentum square of ith particle in system 1 is, [ ] 1 ln p 2 Σ1 i (1) = = T B1, 1 Pramana J. Phys., Vol. 72, No. 2, February 29 317

Vishnu M Bannur an Ramesh Babu Thayyullathil by equipartition theorem (EPT) on system 1. It may be also evaluate by consiering the combine system (1 + 2) as [ ] 1 ln Σ p 2 i (1+2) = = T B, by EPT on system (1+2) together. Similarly, for the other system [ ] 1 ln p 2 Σ2 j (2) = = T B2, 2 by EPT on system 2 which is also equal to [ ] 1 ln Σ p 2 j (1+2) = = T B, by EPT on system (1+2) together. T B1 = T B2 an hence we may call Therefore, once the system is equilibrate T B = ( ln Σ/) 1 (1) as the temperature of the system. Note that it is ifferent from normal efinition of temperature, [ ] 1 ln( Σ/) T S = = Φ 1, where Φ = ( H/ H 2 ), which was foun recently [4] an reformulate an verifie using quartic oscillators as well as Henon Heiles oscillators in ref. [3]. In the limit N both are the same. Let us now iscuss the entropy of our system which is an extensive variable. Following Khinchin [13] for a system with energy E, which consists of two subsystems 1 an 2 we have, Ω(E) = E 1 Ω 1 (E 1 )Ω 2 (E E 1 ), which can be written as Ω(E) = E 1 Ω 1 (E 1 )Ω 2 (E E 1 ). (11) Since the relation given in eq. (11) is a convolution integral, it is convenient to work with Laplace transform of various Ω(E). Defining φ(α) = Ee αe Ω(E) we write φ(α) = φ 1 (α)φ 2 (α) where φ 1 an φ 2 are Laplace transforms of Ω 1 an Ω 2 respectively. Again, following Khinchin, we choose α = θ as the simple root of the equation ln φ(α) α = E. (12) α=θ 318 Pramana J. Phys., Vol. 72, No. 2, February 29

Statistical mechanics for chaotic systems In simple systems like N-quartic oscillators [3] where Σ(E) = CE 3N/4 or ieal gas [17] with Σ(E) = ( ) N V (2πm) 3N/2 h 3 E 3N/2, (3N/2)! etc., this θ is relate to T B via EPT. To see this connection consier systems with Σ(E) = CE l an the corresponing Ω(E) = ClE l 1. Then EPT gives us Since T 1 B φ(α) = = ln Σ = Ω Σ = l E. (13) E e αe ClE l 1 = Cl! α l (14) an using eq. (14) in eq. (12) gives us the relation θ = l E. (15) Comparing eqs (13) an (15) we get the important relation θ = T 1 B. Now the extensive property reas as φ(α) = φ 1 (α)φ 2 (α) φ(t B ) = φ 1 (T B )φ 2 (T B ). In this formalism [13] the entropy is efine as an it becomes S = E θ + ln φ(θ) E T B + ln φ(t B ), (16) S = E 1 + E 2 T B + ln(φ 1 (T B )φ 2 (T B )) = S 1 + S 2, (17) where S 1 an S 2 are entropies of the subsystems. Hence the extensive property of the entropy efine by eq. (16) is very well-establishe. For our examples, an this gives us Finally, it leas to Σ = CE l ln Σ = ln C + l ln E S = ln Σ + ln l! (l ln l l) + constant. S = ln Σ = T 1 B, which is the well-known thermoynamic relation. Pramana J. Phys., Vol. 72, No. 2, February 29 319

Vishnu M Bannur an Ramesh Babu Thayyullathil 4. Canonical ensemble formulation of quartic oscillators In the classical theory we have the canonical partition function Q = p 1 p 2 q 1 q 2 e β[ p 2 1 +p2 2 2 + 1 2 ((1 α)/6 (q4 1 +q4 2 )+ q2 1 q2 2 )], (18) which reuces to Q = (2 π T ) 3/2 2 1 α 2(4 α) K ( ) 2 + α, (19) 4 α where T is the temperature an β = 1/T an K(z) is the complete elliptical integral of the first kin. From the partition function Q one can obtain the ensity of states which may be useful for the stuy of transition from chaos to integrability of a system [18]. For a more general quartic oscillator with N egrees of freeom we have Q N = N p N qe βh, where H = i p 2 i 2 + i,j α ij 2 q2 i q 2 j, which on simplification reuces to Q N = T 3N/4 C N, (2) where C N is inepenent of T. Following the stanar proceure, various thermoynamic quantities may be erive from the Helmholtz free energy, A = T ln Q N = 3N 4 T ln T T ln C N. (21) The entropy S an the average energy U are an S = 3N 4 ln T + 3N 4 + ln C N U = 3 4 NT. Further, the specific heat C V an square of the energy fluctuation E 2 are obtaine as C V = 3 4 N, E 2 U 2 = 1 3N/4. 32 Pramana J. Phys., Vol. 72, No. 2, February 29

Statistical mechanics for chaotic systems It is interesting to note that the energy temperature relation here is the same as that of the microcanonical ensemble formulation, i.e. T 1 B = (3N/4)/E, provie we take the temperature T to be the same as T B. Also note that the temperature T S Φ 1 efine in ref. [4] is obtaine from the efinition Φ = 1 N p N qe β H H Q N H 2, an after some algebra it gives us Φ = β = 1 T, which is also the same as T. This means that T B an T S are the same an equal to T, since T is the temperature of the reservoir with the number of egrees of freeom N an at this limit T B = T S. Thus, for the ynamical evaluation of temperature in canonical ensemble formulation T B may be more preferable, contrary to the observation in ref. [4], since the expression for Φ is more complicate than p 2 i for very large N. 5. Numerical results Numerical results for microcanonical ensemble of QO were reporte earlier [3] where we have showe that the temperatures, which are obtaine by time averaging, agrees with that of the phase-space averaging whenever the system an the initial points are chosen to be in the chaotic region. Here, in the case of canonical ensemble also, the time averages almost agree with that of the phase-space averages for the chaotic QO system. To establish this we have performe a ynamical calculation of various quantities like temperatures (T B an T S ), average energy (U) an specific heat (C V ). In oing this we couple one of the egrees of freeom q 1 of our QO to a thermal bath. An the thermal bath is moelle by Nose Hoover thermostats given by η = q2 1 T 1, (22) where T is the temperature. The final equations of motion which follows from the Hamiltonian in eq. (1) are an q 1 = q 2 = (1 α) q1 3 q 1 q2 2 η q 1 (23) 3 (1 α) q2 3 q 2 q 2 3 1. (24) An we nee the solutions to eqs (22) (24). To check for the accuracy of the numerical metho we may use the following constant of motion of the above set of equations: Pramana J. Phys., Vol. 72, No. 2, February 29 321

Vishnu M Bannur an Ramesh Babu Thayyullathil 2 1.75 1.5 Cv <1.5 1.25 1.75.5 T B2 T B1 U T S <.75 <.5.25 1 1 1 1 1e+6 t Figure 1. Plots of TB of q 1 (TB1) an q 2 (TB2) egrees of freeom, TS, U an CV of chaotic QO system as a function of time. E (p2 1 + p 2 2) 2 + (1 α) 12 + 1 2 q2 1q 2 2 + T η2 2 + T (q 4 1 + q 4 2) t τη(τ). (25) Results are plotte in figure 1 for the thermostat temperature T =.5. Here TB, T S, U an CV are obtaine by time averaging of ynamical quantities. These are plotte as a function of time in figure 1 an asymptotically they all approach the values obtaine from the phase-space averages, i.e., from eq. (21) for N = 2, which are given by T B =.5, T S =.5, U =.75 an C V = 1.5. The QO system is chosen to be highly chaotic with α close to 1. For reasons of numerical stability we took α =.99. As can be seen from the plot, TB1 immeiately approaches.5, the temperature of the thermostats, since it is in irect contact with the reservoir. But the other ynamical quantities TB2, T S, U an CV slowly approach the values.5,.5,.75 an 1.5 respectively in accorance with the ergoic theorem. 6. Conclusions The formulation of statistical mechanics of chaotic systems base on Khinchin s formalism [13] of statistical mechanics for a finite N system is presente. We investigate both microcanonical an canonical ensemble systems of quartic oscillators. The microcanonical ensemble of chaotic systems were stuie an numerically verifie earlier by Berichevsky an Alberti [2], an also by one of us [3]. Here we strengthene the earlier observations an extene the stuy to canonical ensemble 322 Pramana J. Phys., Vol. 72, No. 2, February 29

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