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Soliton Statistical Mechanics and the Thermalisation of Biological Solitons R SOLITON STATISTICAL MECHANICS AND THE THERMALISATION OF BIOLOGICAL SOLITONS R. Bullough, D. Pilling, Yi Cheng, Yu-Zhong Chen, J. Timonen To cite this version: R. Bullough, D. Pilling, Yi Cheng, Yu-Zhong Chen, J. Timonen. SOLITON STATISTICAL ME- CHANICS AND THE THERMALISATION OF BIOLOGICAL SOLITONS. Journal de Physique Colloques, 1989, 50 (C3), pp.C3-41-C3-51. 10.1051/jphyscol:1989306. jpa-00229445 HAL Id: jpa-00229445 https://hal.archives-ouvertes.fr/jpa-00229445 Submitted on 1 Jan 1989 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. JOURNAL DE PHYSIQUE Colloque C3, supplement au ne3, Tome 50, mars 1989 SOLITON STATISTICAL MECHANICS AND THE THERMALISATION OF BIOLOGICAL SOLITONS R.K. BULLOUGH, D.J. PILLING, YI CHENG, W-ZHONG CHEN and J. TIMONEN* Department of Mathematics, ,UMIST, PO Box 88, GB-Manchester, M60 1120, Great-Britain 'Department of Physics, University of Jyvdskyld, SF-40100, ~yvdskyld, Fin1 and Abstract - The calculation of the equilibrium free energy of integrable models like the sine-Gordon and attractive nonlinear Schrodinger models is discussed in the context. of biological molecules like DNA: the thermalisation process (approach to equilibrium) is also discussed. The sine-Gordon model has a "repulsive" form which is the sinh-Gordon model. The approach to equilibrium of the sinh-Gordon model is described in all completeness in terms of a quantum mechanical master equation at finite temperatures. Although the dynamical evolution of the master equation as written is a solved problem, only the equilibrium solution is examined in this paper. The equilibrium free energy is calculated exactly as an integral equation for certain excitation energies. at finite temperatures. Bose-fermi equivalent forms of this integral equation are given. The hose form yields a similar integral equation in classical limit. The iteration of this yields a low temperature asymptotic series for the classical free energy which checks against the result of the transfer integral method (TIM). Results for the zero temperature quantum eigenenergies are found. A further discussion of the dynamics of the approach to thermal equilbriurn is made. 1~- INTRODUCTION It has been suggested (eg./1,2,3/) that the classical sine Gordon model ax, - @tt = m2 sin O (1) is a good model to describe soliton excitations on DNA (here O means a2Q/ax2, etc, the left side is a,, - co-20tt, m is a mass or the wave number mcoft-i~and units are chosen so A = co = 1). Likewise the model may have some relevance to the transmission of soliton-like excitations on the protein a-helix. But here the non-relativistic form of the s-G model which is the non-linear Schrodinger model seems preferable /4,5,6,7/(Davydov's mode1/4,5/ coincides with (2) only in that his soliton is the soliton solution of (2)). Note that the field Q in (1) is real. But in the NLS model (2) Q is complex. Moreover there are actually two NLS models: c is a coupling constant in (2) and c<O is the "attractive NLS"; c>O is the repulsive NLS. The repulsive NLS has no soliton solutions but, like the attractive NLS, it is still an "integrable" model /8,9/. The integrable models are such that they can be solved by the spectral transform (inverse scattering)method /8,9/. In this paper we focus attention on the sine-Gordon model (s-G), equation (1) and its "repulsive" counterpart the sinh-G model which has no solitons. Both are integrable models /8,9/. The soliton solutions of the s-G are the well-known kink and antikink solutions The kink solution (tve sign in (3)) takes @ from zero to 217 as x goes from -oo to +m: the antikink takes O from 2n ko zero. Thus the kink (3) carries a twist of 2n up x with speed V: the antikink (3) carries a twist of -271. The point of such solutions in the biological context is that ax = *2rn(l-~~)-~sech (+rn(x-vt)(l-~Z)-%] , Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1989306 C3-42 JOURNAL DE PHYSIQUE These are narrow soliton pulses of width - (1-V2)* m-I. The velocity V is scaled against co, so as the speed approaches cop the sound velocity ie. V+1, these solitons become very narrow and have large amplitudes. Both kinks and antikinks carry energy ~(1-v2)-%.If we define a momentum pk = MV(~-V~)-* this energy is We encounter this later. The number M is the kink mass: M = 8mya-l in which yo>O is the coupling constant of s-G. We remark further on yo shortly. These remarks are intended to show that the s-G solitons (kink or antikink) are essentially compact localised pulses carrying specific energies (5). In the case of S-GI these solitons are topological solitons and also carry twists of 2n(-2n). Both the pulses, and their twists, can be viewed as "bits" of information. The pulse as a "bit" has been used in a "shift register" with pico-second access /lo/. The soliton solutions of the NLS, equation (2), with c<O are a basis for information transfer in an optical fibre /11/ They figure in the soliton laser/l2/. Neither (1) nor (4) shows evidence of damping. But it is easy to show that even when the s-G system is damped, the solitons remain acceptable solutions (cf. eg. /13/). It is these several facts that could make the soliton an important mode of energy and information transport in biological systems. Certainly the s-G (1) plays an important role in nonlinear physics /14/. For example it describes the excitations of ferromagnetics like CsNiFB /15,16,17/ and antiferromagnetics like TMMC /16,17/. These excitiations are present in thermal equilibrium and apparently govern the neutron scattering cross-sections /16,18/. In biological molecules, the soliton must be subject to thermal agitation. To describe this one can introduce a random force F(t) into equations of motion like (1). This causes diffusion and damping so additionally the left side of (1) gains a term in -K@t associated with the -@tt:K is a damping constant. The random force F(t) appears in a quantum theory developed in Heisenberg representation. Another way to proceed is to work with a quantum theoretical master equation in Schrodinger representation /19/. We do this briefly in this paper (in 5 4). A steady state solution of the master equation is (exp-PH)Z-l: E1 is the temperature T (Boltzmann's constant kg = l), H is the Hamiltonian (operator) and Z is the partition function. The free energy F = -K1ln Z. We can compute Z by the methods of statistical mechanics. It is the statistical mechanics of integrable models like the s-G (I), or the NLS models (2), with which this paper is primarily concerned. To this end we need the complete solution of s-G. In addition to the kinks and anti-kinks there are bound pairs of these, called "breathers" namely @(x,t) = 4 tan-I [tan u sin eI sech +] % = (m sin p)(x-vt)(l-V2)-% eI = (m cos &f)(t-VX) (1-v2)-*. These have energies The rest energy is Mb = 2M sin u. Since 0 < i.d < Hrr, the breather masses form a band in 0 < Mb < 2M. Since m cos u is a frequency (in the chosen units) the breather solution (6) has an internal oscillation sin eI modulated by a sech envelope all under the tan-I function. For small enough !.I d~ (x,t) - 4mi.d sin (m(t-Vx) (I-V~)-~) (8) a harmonic solution of very small amplitude. This is actually a solution of the linearised s-G, the Klein-Gordon (KG) equation However it is well known /9/ that the s-G itself has 3 sorts of solution : the kinks (and antikinks) (3), the breathers (6) and "radiation" approximately described by (8). We shall call this radiation "phonons" in this paper. Thus: the complete solution of s-G (under vanishing boundary conditions at x = w /8,9/)is made up of kinks plus antikinks plus breathers plus phonons. However it is already clear that it may not be very easy to distinguish phonons from small amplitude breathers. This fact plays a role in the statistical mechanics (5 3). The coupling'constant yo introduced by equation (5) allows a continuation from Yo + -Yo. If we set d + eOd(which is actually a canonical transformation /8/) the s-G (1) is Then, as yo -t -Yo, (10) becomes the sinh-G in JE a. Moreover as yo + 0, (10) becomes the KG (9). Thus sinh-G is obtained by continuation in yo from s-G; and both s-G and sinh-G contain KG as yo + 0. Because sinh-G and repulsive NLS have no soliton solutions they might seem to be uninteresting. This is not the case. Certainly the statistical mechanics of these two models is easier than that of s-G or attractive NLS. It is for this reason we shall mostly be concerned with the statistical mechanics of sinh-G in this paper 2 - THE DRESSED AND UNDRESSED NUMBER DENSITITES The number of classical or quantum solitons of the s-G excited at temperature B1 in thermal equilibrium is of physical interest.
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