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Redalyc.Considerations on Undistorted-Progressive X-Waves Brazilian Journal of Physics ISSN: 0103-9733 [email protected] Sociedade Brasileira de Física Brasil Mesquita, Marcus V.; Vasconcellos, Áurea R.; Luzzi, Roberto Considerations on undistorted-progressive X-Waves and davydov solitons, Fröhlich-Bose-Einstein condensation, and Cherenkov-like effect in biosystems Brazilian Journal of Physics, vol. 34, núm. 2A, june, 2004, pp. 489-503 Sociedade Brasileira de Física Sâo Paulo, Brasil Available in: http://www.redalyc.org/articulo.oa?id=46434227 How to cite Complete issue Scientific Information System More information about this article Network of Scientific Journals from Latin America, the Caribbean, Spain and Portugal Journal's homepage in redalyc.org Non-profit academic project, developed under the open access initiative Brazilian Journal of Physics, vol. 34, no. 2A, June, 2004 489 Considerations on Undistorted-Progressive X-Waves and Davydov Solitons, Frohlich-Bose-Einstein¨ Condensation, and Cherenkov-like effect in Biosystems Marcus V. Mesquita,1 Aurea´ R. Vasconcellos,2 and Roberto Luzzi2 1. Instituto de F´ısica, Universidade Federal do Ceara,´ 60455-760, Fortaleza, CE, Brazil 2. Instituto de F´ısica ‘Gleb Wataghin’, Universidade Estadual de Campinas, Unicamp, 13083-970 Campinas, SP, Brazil Received on 3 June, 2003 Research in ultrasonography has evidenced the propagation of a peculiar kind of excitation in fluids. Such excitation, dubbed a X-wave, has characteristics resembling that of a solitary-wave type. We reconsider the problem in a medium consisting of a biological material of the like of ®-helix proteins. It can be shown that in this case is expected an excitation of the Davydov’s solitary wave type, however strongly damped in normal conditions. The case of acetanilide, an organic polymer which resembles biopolymers, is considered, and the infrared spectrum analyzed. Davydov’s soliton is evidenced as a coherent state of polar vibrations. The case of acoustic (sound) vibrations is also considered, where, also, a damped Davydov-like solitary wave may be excited. However, it is shown that when traveling in conditions sufficiently away from equilibrium, more precisely, when the soliton is embeded in the resulting Frohlich-Bose-Einstein¨ condensate, the lifetime of the solitary wave is largely enhanced. Moreover, a soliton moving in bulk with a velocity larger than that of the group velocity of the normal vibrational waves would produce a Cherenkov-like emission of phonons giving rise to the observed X-wave-like pattern. This paper is a modified and extended version of an earlier publication on the subject. 1 Introduction polymers with very large conductivity for, e. g., use in very light, almost two-dimensional (sheets) batteries [7, 8, 9], and Experiments in ultrasonography have evidenced a particular the case of propagation of light in optical fibers [10]. An- kind of wave propagation, dubbed as X-waves [1, 2]. They other example is that of the so-called Davydov’s solitons have the characteristic of propagating without dispersion, [11, 12], which may have a quite relevant role in bioener- and may result of relevance for ultrasound medical imag- getics. ing. Later on, they were ascribed to some kind of the so- Davydov’s theory has received plenty of attention, and a called undistorted propagating waves [3]. They are waves long list of results published up to the first half of 1992 are traveling in a material media, and the reported characteris- discussed in a comprehensive review due to A. C. Scott [13]. tics point to the possibility of they belonging to the category As pointed out in that review, one question concerning of solitary waves. Soliton is the name coined to describe Davydov’s soliton is that of its stability at normal physi- a pulse-like nonlinear wave (the solitary wave referred to ological conditions, that is, the ability of the excitation to above) which emerges from a collision with a similar pulse transport energy (and so information) at long distances in the having unchanged shape or speed. Its relevance in applied living organism, in spite of the relaxation mechanisms that sciences has been described in a 1973 review paper by A. C. are expected to damp it out at very short (micrometers) dis- Scott et al. [4] Because of the above mentioned technologi- tances. On the assumption that the X-waves in ultrasonog- cal/medical relevance we reanalyze the question in the case raphy may belong to the category of solitary waves in the of propagation in biological materials. material media, we consider this question on the basis of a The original category of solitary waves consists in the model in nonequilibrium thermodynamic conditions, since one observed by the Scottish engineer Scott-Russell in Au- any biological system is an open system out of thermal equi- gust 1834 in an English water channel, and reported in a librium. For that purpose we resort to an informational sta- 1844 meeting of the British Society for the Advancement of tistical thermodynamics (see for example [14-19]), based on Science [5, 6]. During the second half of this century many the Nonequilibrium Statistical Operator Method (NESOM) other types of solitary waves have been associated to a num- [20-23]. The NESOM, which provides microscopic foun- ber of physical situations in condensed matter physics. Sev- dations to phenomenological irreversible thermodynamics eral, seemingly, have a fundamental role in important tech- [19], also allows for the construction of a nonlinear gen- nological areas of large relevance for contemporary society. eralized quantum transport theory — a far-reaching gener- Among them we may highlight the case of doped organic alization of the Chapman-Enskog’s and Mori’s methods — 490 Marcus V. Mesquita, Aurea´ R. Vasconcellos, and Roberto Luzzi which describes the evolution of the system at the macro- acoustic-like vibrations, which is described by the so-called scopic level in arbitrary nonequilibrium situations [20,23- Frohlich-Davydov¨ Hamiltonian given in [11] and [27]: This 25], a formalism to be used in what follows. We notice Hamiltonian is given in the so-called Random Wave Ap- that the method has been applied with particular success to proximation, while the full Hamiltonian — to be used in the study of optical and transport properties in solids [26] what follows — is given by and also to modelled biopolymers [27, 28], of the kind we consider here, and to techno-industrial processes [29]. Con- H = H0 + HI = H0S + H0B + HI ; (1) sequently, we do not go over the details of the formalism, just adapting those earlier results for the present discussion, where giving in each case the reference where they are reported. µ ¶ We separate the presentation into two parts: After a general X 1 H = ~! aya + ; (2a) derivation of the equations of evolution, a first analysis is 0S ~q ~q ~q 2 concentrated on the case of polar vibrations (with frequen- ~q cies in the infrared region of the spectrum), for which are available experimental results, particularly the case of ac- X 1 etanilide, which we analyze, comparing theory and experi- H = ~­ (byb + ) ; (2b) ment. This is done in subsection 2.1, while in subsection 2.2 0B ~q ~q ~q 2 ~q we deal with the case of longitudinal acoustic vibrations. As stated, we first consider a system where modes of po- lar vibrations are excited by a continuous supply of energy. X X These polar modes are coupled through a nonlinear kinet- y (1) y HI = Z~q' a + V a~q b~q b ics with a bath consisting of a continuous medium modelled ~q ~q ~q1~q2 1 2 ~q1+~q2 by a system of acoustic-like vibrations. The equations of ~q ~q1~q2 evolution for the population of the vibrational modes are de- X X rived resorting to the nonlinear quantum kinetic theory that + V (1) a by b + V (1) a b b ~q1~q2 ~q1 ~q2 ¡~q1+~q2 ~q1~q2 ~q1 ~q2 ¡~q1¡~q2 the NESOM provides. This corresponds to the description ~q1~q2 ~q1~q2 of polar vibrations of the CO-stretching type (Amide-I) in, X X for example, ®-helix proteins [11-13]. Experiments, of the (1) y y (2) y + V a~q b b + V a~q a~q b class of Raman or neutron scattering or radiation absorp- ~q1~q2 1 ~q2 ~q1¡~q2 ~q1~q2 1 2 ~q1+~q2 ~q ~q ~q ~q tion, in active biological systems are particularly difficult. 1 2 1 2 X X For that reason, the comparison of the theory of next sec- (2) (2) y + V a~q a~q b¡~q ¡~q + V a a~q b~q ¡~q tion with experiment is done in the case of the organic poly- ~q1~q2 1 2 1 2 ~q1~q2 ~q1 2 1 2 mer acetanilide, which constitutes a good mimic of biopoly- ~q1~q2 ~q1~q2 mers [30]. The infrared spectrum in the frequency region X + V (2) a ay by + H.c. (2c) corresponding to the CO-stretching oscillation is derived, ~q1~q2 ~q1 ~q2 ~q1¡~q2 and compare very well with the experimental measurements, ~q1~q2 confirming the presence of Davydov’s soliton. Finally, we comment on the possibly very large increase of the lifetime Figure 1 in [27] describes a particular biological system of Davydov’s soliton when propagating in an open medium and the mechanical analog we are using. The Hamiltonian sufficiently far from equilibrium. consists of the energy of the free subsystems, namely, that In subsection 2.2, we consider a system of longitudinal of the free vibrations, with !~q being their frequency disper- acoustic vibrations in interaction with the above said ther- sion relation (~q is a wave-vector running over the reciprocal- mal bath. We find a behavior of the acoustic modes quite space Brillouin zone), and that of the thermal bath composed similar to the one evidenced for the optical modes, namely, by oscillations with frequency dispersion relation ­~q, with existence of the solitary-wave excitation, damped near equi- a Debye cut-off frequency ­D.
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