Slow (logarithmic) relaxation from a highly excited state is studied in a Hamiltonian system with many degrees of freedom. The relaxation time is shown to increase as the exponential of the square root of the energy of excitation, in agreement with the Boltzmann-Jeans conjecture, while it is found to be inversely proportional to residual Kolmogorov-Sinai entropy, introduced in this Letter. The increase of the thermodynamic entropy through this relaxation process is found to be proportional to this quantity.

pacs:
05.45.Jn 05.70.Ln 05.45-a 87.10.+e

Study of the relaxation from highly excited states is important not only in physico-chemical systems but also in biological systems. It has been reported that excitations of some protein molecules are maintained over anomalously long time spans [1]. Such behavior is relevant to enzymatic reactions, and in particular to their biological functions, because these molecules must maintain their excited state (brought about, say, by ATP) to be able to proceed from one process to the next.

Typically, a Hamiltonian system with a sufficiently large number of degrees of freedom relaxes to equilibrium. This relaxation process, however, is not necessarily fast. For instance, some Hamiltonian systems can exhibit logarithmically slow relaxation from excited states, called Boltzmann-Jeans conjecture (BJC), which was first noted by Boltzmann[2], explored by Jeans[3], Landau and Teller[4], and then has been studied in terms of nonlinear dynamics [5]. In the BJ conjecture, the relaxation to equilibrium is required, but existence of chaos is not explicitly assumed. On the other hand, irreversible relaxation is often studied in relationship with chaos. In the present paper we study slow relaxation of a type in agreement with the BJC, by using a Hamiltonian dynamical system with a large number of degrees of freedom, and explore its possible relationship with chaos.

We consider dynamics with the Hamiltonian

H=K+V=∑i=1Npi22+∑i,j=1NV(θi,θj)ݐ»ݐ¾ݑ‰superscriptsubscriptݑ–1ݑsuperscriptsubscriptݑݑ–22superscriptsubscriptݑ–ݑ—1ݑݑ‰subscriptݜƒݑ–subscriptݜƒݑ—H=K+V=\sum\limits_i=1^Np_i^2\over 2+\sum_i,j=1^NV(\theta_i,% \theta_j)italic_H = italic_K + italic_V = ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT divide start_ARG italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 2 end_ARG + ∑ start_POSTSUBSCRIPT italic_i , italic_j = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT italic_V ( italic_θ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_θ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) (1)
describing a set of pendula, where pisubscriptݑݑ–p_iitalic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT is the momentum of the iݑ–iitalic_i-th pendulum. The potential Vݑ‰Vitalic_V is chosen so that each pair of pendula interacts only through their phase difference [6, 7]:

V(θi,θj)=12(2π)2N1-cos(2π(θi-θj)).ݑ‰subscriptݜƒݑ–subscriptݜƒݑ—12superscript2ݜ‹2ݑ12ݜ‹subscriptݜƒݑ–subscriptݜƒݑ—V(\theta_i,\theta_j)=1\over2(2\pi)^2N\1-\cos(2\pi(\theta_i-\theta% _j))\.italic_V ( italic_θ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_θ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) = divide start_ARG 1 end_ARG start_ARG 2 ( 2 italic_π ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_N end_ARG 1 - roman_cos ( 2 italic_π ( italic_θ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - italic_θ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) ) . (2)
Hence, the evolution equations for the momentum pisubscriptݑݑ–p_iitalic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT and the phase θisubscriptݜƒݑ–\theta_iitalic_θ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT are given by

p˙j=12πN∑i=1Nsin(2π(θi-θj)),θ˙j=pj.formulae-sequencesubscript˙ݑݑ—12ݜ‹ݑsuperscriptsubscriptݑ–1ݑ2ݜ‹subscriptݜƒݑ–subscriptݜƒݑ—subscript˙ݜƒݑ—subscriptݑݑ—\dotp_j=1\over2\pi N\sum_i=1^N\sin(2\pi(\theta_i-\theta_j)),{% }{}{}\dot\theta_j=p_j.over˙ start_ARG italic_p end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG 2 italic_π italic_N end_ARG ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT roman_sin ( 2 italic_π ( italic_θ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - italic_θ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) ) , over˙ start_ARG italic_θ end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = italic_p start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT . (3)
This form of the interaction is chosen so that there is an attractive force tending to align the phases of all the pendula. The thermodynamic properties of this model in the N→∞→ݑN\rightarrow\inftyitalic_N → ∞, as well as the finite-size effect, are investigated in Ref.[7, 8]. Here we mainly discuss the case with N=10ݑ10N=10italic_N = 10, which is sufficiently large to exhibit the thermodynamic properties, and the results to be discussed are valid for a larger system. The temperature of the system can be defined as T≡⟨2K⟩/Nݑ‡delimited-⟨⟩2ݐ¾ݑT\equiv\langle 2K\rangle/Nitalic_T ≡ ⟨ 2 italic_K ⟩ / italic_N, which is a monotonically increasing function of the total energy Eݐ¸Eitalic_E.

As studied in Refs.[6, 7, 8, 9], each pendulum in this system exhibits small-amplitude vibration when the total energy is small, while, as the energy is increased, some pendula begin to display rotational behavior over many cycles. The relaxation of the rapidly rotating pendula into a vibrating assembly is rather slow, since their average interaction with the assembly almost cancels out over the slow time scale of the assembly, due to the rapid rotation. Here we concentrate on such slow relaxation of the rotational mode to the vibrational modes.

The slow relaxation from a highly excited state is investigated systematically in the situation that the excited state is prepared by applying an instantaneous kick to a certain pendulum in the system. An example of the relaxation process following this kick is depicted in Fig.1, where it is seen that the kicked pendulum continues to rotate in isolation, maintaining a large energy. The relaxation time tRsubscriptݑ¡ݑ…t_Ritalic_t start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT is defined as the interval of time from the kick to the point of which kinetic energy of the kicked pendulum first becomes smaller than K/Nݐ¾ݑK/Nitalic_K / italic_N.

The relaxation process depends on the energy E0subscriptݐ¸0E_0italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT imported by the kick and also on the state of the other N-1ݑ1N-1italic_N - 1 pendula. Hereafter we call this assembly the ‘bulk’. The bulk state is parameterized by the total energy Eݐ¸Eitalic_E (or the temperature Tݑ‡Titalic_T) of the system before the kick [10].

The mean relaxation time ⟨tR⟩delimited-⟨⟩subscriptݑ¡ݑ…\langle t_R\rangle⟨ italic_t start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT ⟩ over an ensemble of initial conditions was computed by fixing Eݐ¸Eitalic_E and varying E0subscriptݐ¸0E_0italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT. As shown in Fig.2, the relaxation time increases exponentially with the kicked energy E0subscriptݐ¸0E_0italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT [11] as

⟨tR⟩∝exp(αE0),proportional-todelimited-⟨⟩subscriptݑ¡ݑ…ݛ¼subscriptݐ¸0\langle t_R\rangle\propto\exp(\alpha\sqrtE_0),⟨ italic_t start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT ⟩ ∝ roman_exp ( italic_α square-root start_ARG italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG ) , (4)
for sufficiently large values of E0subscriptݐ¸0E_0italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT. Here ⟨⋅⟩delimited-⟨⟩⋅\langle\cdot\rangle⟨ ⋅ ⟩ represents an ensemble average and αݛ¼\alphaitalic_α is a constant. When the value of E0subscriptݐ¸0E_0italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT is fixed, ⟨tR⟩delimited-⟨⟩subscriptݑ¡ݑ…\langle t_R\rangle⟨ italic_t start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT ⟩ decreases as a function of Eݐ¸Eitalic_E, as is seen in the figure.

The above exponential form of the relaxation agrees with that given by the BJC, which describes the relaxation time for a system consisting of two parts with very different frequencies as an exponential of the ratio of the two frequencies. Assuming the separation of time scales of the kicked pendulum and the bulk pendula, this exponential growth can be obtained through the Fourier analysis of the interaction term, as shown by Landau and Teller [4] (see also [5]).

In this argument, chaotic dynamics are not explicitly included, while chaos is often relevant to the irreversible relaxation to equilibrium. Actually, we have observed the exponential law (4) in numerical studies clearly, as long as the Hamiltonian dynamics for the given energy Eݐ¸Eitalic_E uniformly exhibits chaotic behavior without remnants of KAM tori. In this regime, ⟨tR⟩delimited-⟨⟩subscriptݑ¡ݑ…\langle t_R\rangle⟨ italic_t start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT ⟩ decreases with the energy Eݐ¸Eitalic_E, as shown in the inset of Fig.2. On the other hand, in the regime of the small values of Eݐ¸Eitalic_E, the relaxation process highly depends on the initial condition and it is difficult to define average relaxation time. The hindrance of relaxation is associated with the weakness of chaos, since the relaxation of the rotating pendulum stops when the dynamics of the bulk is trapped to the vicinity of remnants of tori. Now, we study a possible relationship between the relaxation time and the chaotic dynamics of the bulk.

First let us consider the nature of the dynamics in the limit E0→∞→subscriptݐ¸0E_0\rightarrow\inftyitalic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT → ∞. In this case, the kicked pendulum is completely free from the bulk of vibrating pendula, whose collective motion is simply that determined by the Hamiltonian of its (N-1)ݑ1(N-1)( italic_N - 1 ) pendula. At a finite value of E0subscriptݐ¸0E_0italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, the dynamics of the bulk deviate from those in the E0→∞→subscriptݐ¸0E_0\rightarrow\inftyitalic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT → ∞ limit, due to the interaction with the kicked pendulum. In order to quantify such deviation, we choose bulk states with energies distributed around a given Eݐ¸Eitalic_E, and compute the Lyapunov spectrum λi(E0)subscriptݜ†ݑ–subscriptݐ¸0\lambda_i(E_0)italic_λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) for various values of E0subscriptݐ¸0E_0italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT. Note that, in the limit E0→∞→subscriptݐ¸0E_0\rightarrow\inftyitalic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT → ∞, λN-2=0subscriptݜ†ݑ20\lambda_N-2=0italic_λ start_POSTSUBSCRIPT italic_N - 2 end_POSTSUBSCRIPT = 0 corresponding to the free rotation of the kicked pendulum, while λN-1=λN=0subscriptݜ†ݑ1subscriptݜ†ݑ0\lambda_N-1=\lambda_N=0italic_λ start_POSTSUBSCRIPT italic_N - 1 end_POSTSUBSCRIPT = italic_λ start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT = 0 always holds due to the conservation of the total momentum and energy.

Since the kinetic energy Krsubscriptݐ¾ݑŸK_ritalic_K start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT of the kicked pendulum fluctuates and relaxes, λi(E0)subscriptݜ†ݑ–subscriptݐ¸0\lambda_i(E_0)italic_λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) is computed in the case that Krsubscriptݐ¾ݑŸK_ritalic_K start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT satisfies |E0-Kr|<δsubscriptݐ¸0subscriptݐ¾ݑŸݛ¿|E_0-K_r|| italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT - italic_K start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT |
The values of the computed Lyapunov exponents increase as E0subscriptݐ¸0E_0italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT decreases. The increased part of the Lyapunov exponents from those at E0→∞→subscriptݐ¸0E_0\rightarrow\inftyitalic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT → ∞ limit characterizes the amplification of the chaotic instability due to the interaction between the kicked pendulum and the bulk. With this in mind, we define the residual Lyapunov spectra by

Δλi(E0)≡λi(E0)-limE0→∞λi(E0).Δsubscriptݜ†ݑ–subscriptݐ¸0subscriptݜ†ݑ–subscriptݐ¸0subscript→subscriptݐ¸0subscriptݜ†ݑ–subscriptݐ¸0\Delta\lambda_i(E_0)\equiv\lambda_i(E_0)-\lim_E_0\rightarrow\infty% \lambda_i(E_0).roman_Δ italic_λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) ≡ italic_λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) - roman_lim start_POSTSUBSCRIPT italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT → ∞ end_POSTSUBSCRIPT italic_λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) . (5)
¿From detailed numerical experiments, we find that the two scaling relationships;

λN-2(E0)=ΔλN-2(E0)∝exp(-β1E0),subscriptݜ†ݑ2subscriptݐ¸0Δsubscriptݜ†ݑ2subscriptݐ¸0proportional-tosubscriptݛ½1subscriptݐ¸0\displaystyle\lambda_N-2(E_0)=\Delta\lambda_N-2(E_0)\propto\exp(-\beta% _1\sqrtE_0),italic_λ start_POSTSUBSCRIPT italic_N - 2 end_POSTSUBSCRIPT ( italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) = roman_Δ italic_λ start_POSTSUBSCRIPT italic_N - 2 end_POSTSUBSCRIPT ( italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) ∝ roman_exp ( - italic_β start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT square-root start_ARG italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG ) , (6)

Δh(E0)≡∑i=1N-3Δλi∝exp(-β2E0)Δℎsubscriptݐ¸0superscriptsubscriptݑ–1ݑ3Δsubscriptݜ†ݑ–proportional-tosubscriptݛ½2subscriptݐ¸0\displaystyle\Delta h(E_0)\equiv\sum_i=1^N-3\Delta\lambda_i\propto\exp% (-\beta_2\sqrtE_0)roman_Δ italic_h ( italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) ≡ ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N - 3 end_POSTSUPERSCRIPT roman_Δ italic_λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∝ roman_exp ( - italic_β start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT square-root start_ARG italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG ) (7)
hold for sufficiently large values of E0subscriptݐ¸0E_0italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT. Here β1subscriptݛ½1\beta_1italic_β start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT is smaller than β2subscriptݛ½2\beta_2italic_β start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT, as shown in Fig.3(a).

Since λN-2=0subscriptݜ†ݑ20\lambda_N-2=0italic_λ start_POSTSUBSCRIPT italic_N - 2 end_POSTSUBSCRIPT = 0 in the limit E0→∞→subscriptݐ¸0E_0\rightarrow\inftyitalic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT → ∞, the scaling of λN-2subscriptݜ†ݑ2\lambda_N-2italic_λ start_POSTSUBSCRIPT italic_N - 2 end_POSTSUBSCRIPT represents the decrease in the degree of chaos of the kicked pendulum. The sum of the remaining positive Lyapunov exponents, then, is expected to correspond to the Kolmogorov-Sinai (KS) entropy of the bulk, and the increase in the degree of chaos of the bulk due to the interaction with the kicked pendulum is given by ΔhΔℎ\Delta hroman_Δ italic_h. (Note that the increase of Lyapunov exponents is not due to the increase of the average velocity in the bulk. Indeed, we have numerically confirmed that the temperature of the bulk for finite E0subscriptݐ¸0E_0italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT is not shifted from that in the limit E0→∞→subscriptݐ¸0E_0\rightarrow\inftyitalic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT → ∞.)

Since ΔhΔℎ\Delta hroman_Δ italic_h is associated with the dynamics of the bulk and how it is affected by the energy absorbed from the kicked pendulum, we expect that it is related to the relaxation time ⟨tR⟩delimited-⟨⟩subscriptݑ¡ݑ…\langle t_R\rangle⟨ italic_t start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT ⟩ [13]. Indeed αݛ¼\alphaitalic_α in Eq.(4) and β2subscriptݛ½2\beta_2italic_β start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT in Eq.(7) appears to be equal from our numerical results. The relationship between ⟨tR⟩delimited-⟨⟩subscriptݑ¡ݑ…\langle t_R\rangle⟨ italic_t start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT ⟩ and ΔhΔℎ\Delta hroman_Δ italic_h is plotted in Fig.3(b), which supports the relationship

Δh∝⟨tR⟩-1.proportional-toΔℎsuperscriptdelimited-⟨⟩subscriptݑ¡ݑ…1\Delta h\propto\langle t_R\rangle^-1.roman_Δ italic_h ∝ ⟨ italic_t start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT ⟩ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT . (8)
Thus, we find that the degree of chaos is closely related to the relaxation time scale.

The relationship between the relaxation and the residual KS entropy ΔhΔℎ\Delta hroman_Δ italic_h can be roughly explained as follows: In a Hamiltonian system, for every orbit, there exists a time-reversed orbit, obtained by changing the sign of all pisubscriptݑݑ–p_iitalic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT. However, because of the interaction with the kicked element, to obtain the time-reversed orbit of the bulk, a slight modification of each pendulum, in addition to the simple reversed of each of their momenta, pi→-pi→subscriptݑݑ–subscriptݑݑ–p_i\rightarrow-p_iitalic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → - italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT, is needed. Since the KS entropy hℎhitalic_h is a measure of the time rate of loss of the information concerning the initial conditions as an orbit evolves, the measure of orbits that can be considered the reversed orbit of some given orbit over a time tݑ¡titalic_t decreases with exp(-ht)ℎݑ¡\exp(-ht)roman_exp ( - italic_h italic_t ). Therefore, the time scale for the irreversible relaxation is proportional to the inverse of the KS entropy difference between the original and the reversed orbits. Hence, the relationship Eq.(8) between the time scale for the relaxation and the residual KS entropy is reasonable.

Next, we study dynamics on the long time scale of the order of ⟨tR⟩delimited-⟨⟩subscriptݑ¡ݑ…\langle t_R\rangle⟨ italic_t start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT ⟩. To compare the relaxation from various kicked energies E0subscriptݐ¸0E_0italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, it is convenient to use the scaled time τ=t/⟨tR⟩ݜݑ¡delimited-⟨⟩subscriptݑ¡ݑ…\tau=t/\langle t_R\rangleitalic_τ = italic_t / ⟨ italic_t start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT ⟩, where ⟨tR⟩delimited-⟨⟩subscriptݑ¡ݑ…\langle t_R\rangle⟨ italic_t start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT ⟩ is a function of E0subscriptݐ¸0E_0italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT. We note that the time scale of relaxation decreases as the energy of the kicked pendulum, Krsubscriptݐ¾ݑŸK_ritalic_K start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT, decreases. Then the above definition of τݜ\tauitalic_τ implies τ=t/tR(K~r)ݜݑ¡subscriptݑ¡ݑ…subscript~ݐ¾ݑŸ\tau=t/t_R(\tildeK_r)italic_τ = italic_t / italic_t start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT ( over~ start_ARG italic_K end_ARG start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ) along the relaxation, where K~rsubscript~ݐ¾ݑŸ\tildeK_rover~ start_ARG italic_K end_ARG start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT is the coarse-grained energy of the kicked pendulum over a long time scale and tR∝exp(αK~r)proportional-tosubscriptݑ¡ݑ…ݛ¼subscript~ݐ¾ݑŸt_R\propto\exp(\alpha\sqrt\tildeK_r)italic_t start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT ∝ roman_exp ( italic_α square-root start_ARG over~ start_ARG italic_K end_ARG start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT end_ARG ).

Now, we derive an equation describing the relaxation of K~rsubscript~ݐ¾ݑŸ\tildeK_rover~ start_ARG italic_K end_ARG start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT on the time scale of τݜ\tauitalic_τ. We have computed the difference δKrݛ¿subscriptݐ¾ݑŸ\delta K_ritalic_δ italic_K start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT over the time steps δτݛ¿ݜ\delta\tauitalic_δ italic_τ for many samples of orbits starting from a given Krsubscriptݐ¾ݑŸK_ritalic_K start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT. As shown in Fig.4, the data are satisfactory fit by ⟨δKr⟩δτ=-C,delimited-⟨⟩ݛ¿subscriptݐ¾ݑŸݛ¿ݜݐ¶\frac\langle\delta K_r\rangle\delta\tau=-C,divide start_ARG ⟨ italic_δ italic_K start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ⟩ end_ARG start_ARG italic_δ italic_τ end_ARG = - italic_C , with a constant Cݐ¶Citalic_C (≃0.035similar-to-or-equalsabsent0.035\simeq 0.035≃ 0.035) for various values of Krsubscriptݐ¾ݑŸK_ritalic_K start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT and Eݐ¸Eitalic_E. Then the coarse-grained equation for the relaxation is obtained as dKr/dτ=-C.ݑ‘subscriptݐ¾ݑŸݑ‘ݜݐ¶dK_r/d\tau=-C.italic_d italic_K start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT / italic_d italic_τ = - italic_C . Noting that the interaction energy between the bulk and the kicked pendulum is tiny and recalling the conservation of energy, we get dE/dτ=C.ݑ‘ݐ¸ݑ‘ݜݐ¶dE/d\tau=C.italic_d italic_E / italic_d italic_τ = italic_C . Thus, with regard to the relaxation process, the difference in the details of the dynamical properties of the system under different conditions are eliminated by considering the scaled time τݜ\tauitalic_τ, while the dynamics and properties of the bulk seem to be strongly dependent on the value of Eݐ¸Eitalic_E. ¿From the above equation, for the original time variable tݑ¡titalic_t, Krsubscriptݐ¾ݑŸK_ritalic_K start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT decays in the logarithmic time scale.

Now it is possible to consider the entropy increase of the bulk occurring during the relaxation. Since we have assumed that the slow change of the bulk part can be described by the slow change of thermodynamic quantities, the increase of the entropy of the bulk is estimated as

dSdτ=dSdEdEdτ=CT,ݑ‘ݑ†ݑ‘ݜݑ‘ݑ†ݑ‘ݐ¸ݑ‘ݐ¸ݑ‘ݜݐ¶ݑ‡\fracdSd\tau=\fracdSdE\fracdEd\tau=\fracCT,divide start_ARG italic_d italic_S end_ARG start_ARG italic_d italic_τ end_ARG = divide start_ARG italic_d italic_S end_ARG start_ARG italic_d italic_E end_ARG divide start_ARG italic_d italic_E end_ARG start_ARG italic_d italic_τ end_ARG = divide start_ARG italic_C end_ARG start_ARG italic_T end_ARG , (9)
where Tݑ‡Titalic_T is the temperature of the bulk. This relation is obtained by noting that the process here is sufficiently slow to define these thermodynamic quantities on a coarse-grained time scale.

Since τ=t/tR∝Δhtݜݑ¡subscriptݑ¡ݑ…proportional-toΔℎݑ¡\tau=t/t_R\propto\Delta h{}titalic_τ = italic_t / italic_t start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT ∝ roman_Δ italic_h italic_t, by Eq.(8), the relation

dSdt∝Δhproportional-toݑ‘ݑ†ݑ‘ݑ¡Δℎ\fracdSdt\propto\Delta hdivide start_ARG italic_d italic_S end_ARG start_ARG italic_d italic_t end_ARG ∝ roman_Δ italic_h (10)
for the relaxation process is obtained from Eq.(9). The proportionality in Eq. (10) relates two kinds of directed motion towards equilibrium: ΔhΔℎ\Delta hroman_Δ italic_h corresponds to the amplification of chaos in the equi-energy surface of the bulk, whereas dS/dtݑ‘ݑ†ݑ‘ݑ¡dS/dtitalic_d italic_S / italic_d italic_t represents the growth of this equi-energy surface towards equilibrium. It should also be noted that this relationship holds not near the equilibrium state of the total system but for the nonlinear relaxation process from a highly excited state.

The relationship between chaotic dynamics and irreversibility has been extensively studied [13, 14, 15]. A relationship between the thermodynamic entropy and the irreversible information loss was proposed by Sasa and Komatsu in the case that an external operation causes a transition from one equilibrium state to another. We believe that our relation between the residual KS entropy and the thermodynamic entropy is related to it, although ours applies to the spontaneous relaxation.

To sum up, we have obtained the following relationship between the relaxation time, tRsubscriptݑ¡ݑ…t_Ritalic_t start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT, and the energy supplied by an external kick, E0subscriptݐ¸0E_0italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT: tR∝exp(αE0)proportional-tosubscriptݑ¡ݑ…ݛ¼subscriptݐ¸0t_R\propto\exp(\alpha\sqrtE_0)italic_t start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT ∝ roman_exp ( italic_α square-root start_ARG italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG ). Then, this relaxation time tRsubscriptݑ¡ݑ…t_Ritalic_t start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT is found to be proportional to the inverse of the residual KS entropy, that is, the difference between the KS entropy of the bulk interacting with the kicked element and that of the isolated bulk. By rescaling time by tRsubscriptݑ¡ݑ…t_Ritalic_t start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT, the excited energy is found to relax at a rate that is independent of the energy of the kicked element and the bulk. Finally, we find evidence that the rate of entropy generation is proportional to the residual KS entropy.

Although these three relationships were found through numerical study of the Hamiltonian (1), we expect that they hold generally for relaxation processes in Hamiltonian systems under the condition that in the limit of high excitation, the interaction between the excited elements and the unexcited bulk vanishes. Indeed, we have numerically found that Eqs.(4) and (8) are valid also for a coupled pendulum model with the nearest-neighbor coupling on a square lattice. It is important to examine the universality of the three relationships we have found, and also to study their relevance in regard to intra-molecular relaxation processes, including application to the energy transduction carried out by biological molecular motors.

The authors are grateful to S. Sasa and T. S. Komatsu for stimulating discussions. This research was supported by Grants-in-Aids for Scientific Research from JSPS and the REIMEI Research Resources of JAERI.

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