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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�
JOURNAL DE PHYSIQUE
C o l l o q u e C3, s u p p l e m e n t au n e 3 , 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*
D e p a r t m e n t of M a t h e m a t i c s , ,UMIST, PO Box 88, GB-Manchester, M60 1120, G r e a t - B r i t a i n
' D e p a r t m e n t of P h y s i c s , U n i v e r s i t y of J y v d s k y l d , SF-40100, ~ y v d s k y l d , Fin1 a n d
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 a t 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 i s calculated exactly a s an integral equation for certain excitation energies. a t finite temperatures.
Bose-fermi equivalent forms of this integral equation a r e 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 a r e found. A f u r t h e r discussion of the dynamics of the approach to thermal equilbriurn is made.
1~
-
INTRODUCTIONI t 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 m c o f t - i ~ a n d 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 t h e non-relativistic form of the s-G model which is the non-linear Schrodinger modelseems 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 i s complex. Moreover there a r e 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 t h e attractive NLS, i t is still an "integrable" model /8,9/. The integrable models a r e such that they can be solved by t h e 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 t h e s-G a r e the well-known kink and antikink solutions
The kink solution ( t v e sign in (3)) takes @ from zero t o 217 as x goes from -oo to +m: the antikink takes O from 2n ko zero. Thus t h e kink (3) carries a twist of 2n up x with speed V:
t h e antikink (3) carries a twist of -271.
The point of such solutions i n the biological context i s that
ax =
* 2 r n ( l - ~ ~ ) - ~ sech (+rn(x-vt)(l-~Z)-%] ,Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1989306
These a r e narrow soliton pulses of width
-
(1-V2)* m-I. The velocity V is scaled against co, so a s 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=
M V ( ~ - V ~ ) - * this energy isWe 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 a r e intended to show that the s-G solitons (kink or antikink) a r e essentially compact localised pulses carrying specific energies (5). In the case of S-GI these solitons a r e topological solitons and also carry twists of 2n(-2n). Both the pulses, and their twists, can be viewed a s "bits" of information. The pulse a s 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 a r e a basis for information transfer in an optical fibre /11/ They figure in t h e soliton laser/l2/.
Neither (1) nor (4) shows evidence of damping. But it is easy to show that even when the s-G system i s damped, the solitons remain acceptable solutions (cf. eg. /13/). I t i s 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 i t describes the excitations of ferromagnetics like CsNiFB /15,16,17/ and antiferromagnetics like TMMC /16,17/. These excitiations a r e 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 i s 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 i s (exp-PH)Z-l: E1 is the temperature T (Boltzmann's constant k g
=
l), H i s the Hamiltonian (operator) and Z i s the partition function.The free energy F
=
-K1ln Z. We can compute Z by the methods of statistical mechanics. I t is the statistical mechanics of integrable models like the s-G ( I ) , or the NLS models (2), with which this paper i s 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
sineI
sech +]%
= ( m s i n p)(x-vt)(l-V2)-%eI
= (m cos &f) (t-VX) (1-v2)-*.These have energies
The r e s t energy i s Mb
=
2M sinu.
Since 0<
i.d<
Hrr, the breather masses form a band in 0<
Mb
<
2M. Since m cosu
is a frequency (in the chosen units) the breather solution (6) has an internal oscillation sineI
modulated by a sech envelope all under the tan-I function. For small enough !.Id~ (x,t)
-
4mi.d sin (m(t-Vx) ( I - V ~ ) - ~ ) (8)a harmonic solution of very small amplitude. This i s actually a solution of the linearised s-G, the Klein-Gordon (KG) equation
However i t i s 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 a t x
=
w /8,9/)is made up of kinks plus antikinks plus breathers plus phonons. However it is already clear that i t may not be v e r y 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 s e t d + e O d (which i s actually a canonical transformation / 8 / ) the s-G (1) is
Then, a s yo -t -Yo, (10) becomes t h e sinh-G in
JE a.
Moreover a s yo + 0, (10) becomes t h e KG (9). Thus sinh-G is obtained by continuation in yo from s-G; and both s-G and sinh-G contain KG a s yo + 0.Because sinh-G and repulsive NLS have no soliton solutions they might seem to be uninteresting. This is not t h e case. Certainly the statistical mechanics of these two models i s easier than that of s-G or attractive NLS. I t 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 DENSITITESThe number of classical or quantum solitons of the s-G excited a t temperature B1 in thermal equilibrium is of physical interest. If one computes the 1-particle partition function
m +H L
--Q) -H L
for kinks and uses nk
=
L-I{a(B1
In Z)/au],,=o (D is here a chemical potential) one finds /20/The series on the right is a n asymptotic expansion valid for small (MP)-I (low T). From ZE for antikinks one finds nk
=
ny; while t h e breather density i s similarly /20/r
1In (12) and (13) both Kn and I n a r e modified Bessel functions a s in 1203. One finds t h e f r e e energy per unit length a s
and FKG i s the f r e e KG contribution
FKG
=
lim k l a - 1 ['ln(n&-I)+
H ma-
11. (15) a+OI t is no s u r p r i s e that (15) diverges: the KG, equation (9), is in effect, a bunch of harmonic oscillators with dispersion
(Fourier transform (9) to k-space). Classically t h e divergence (15) i s then t h e classical ultra-violet divergence. This does not arise a s such in t h e quantum case (R. K. Bullough, Yu-zhong Chen, and J. Timonen, in preparation): this is because FKG arises from t h e phonons and there a r e no phonons in the quantum case of s-G (/21/ and see below). From (12), (13), j14), (15) we t h u s have
with FKG given by (15). Although (15) diverges one finds the form (15) with a>O comes from the KG (9) alone by using the cut-off J k ( < na-l in k-space. But, a s was partly recognized a t the time /16/, the precise form of (17) i s misleading. A deeper calculation (by t h e Transfer Integral Method (TIM) /8,22/ or by the new methods /22/ sketched again in this paper) shows t h a t
plus terms in e-213M, e-3m, etc. multiplying further asymptotic series. The extra factor 2(mP), the coefficient -3% in the kink-antikink series, and t h e term in (MP)-I in the breather series in (18) a r e due to "dressing" effects from the phonons in the problem: the new series multiplied by e - Z m , a r e "multisoliton" (multikink o r multiantikink) effects.
An important conclusion emerges from the comparision of (17) and (18). Suppose the s-G kink i s a good model of a biological soliton on DNA (say), /1,2,3/: i t will be subject to thermal agitation in vivo and, in the absence of other forces, a random force F(t) describing i t s Brownian motion under this agitation would ultimately drive the soliton into thermal equilibrium with a free energy FL-*
=
- n k r l with nk given by (12). However, a s such solitons accumulate, they must eventually thermalise still further reaching a total free energy density (18) in final thermal equilibrium. Of course i t is not clear that the thermalisation sequence necessarily passes through (17). But certainly (18) is t h e equilibrium free energy while -Pnk, nk from (12). is the equilibrium for a single kink undergoing Brownian motion.The dressing process to (18) will take time and a kink originally travelling with energy (4)
>>
P-I will slow down on some time scale K ~ say (Kk i s a damping constant for this dressing - ~
process). To calculate ~k we need to describe this whole thermalisation process much more completely. Such a dynamical calculation i s still to be done but we sketch some ideas in this connection in 54 next.
The conclusions so f a r are:-
(1) In the absence of other dynamical forces, a biological soliton (kink), will thermalize on some time scale K ~ eventually reaching thermal equilibrium; - ~
(2) In thermal equilibrium the number densities and free energies of classical kinks and antikinks a r e strongly dressed by the phonons (although l3-I
-
300'K we can expect mP i s small so FL-I for kinks in (18) i s increased by this dressing process);(3) The number densities and free energies of the breathers a r e also substantially changed: we can show that the classical breathers actually become large amplitude phonons /21/: these a r e interacting phonons which d r e s s the kinks and antikinks: they give rise to the phonon series in (18) while classical breathers actually disappear /21/.
( 4 ) Free solitons (kinks, antikinks) thermalise to dressed solitons in equilibrium: free breathers thermalise to phonons and otherwise disappear.
(5) These remarks refer to classical kinks and antikinks and breathers: the quantum s-G behaves quite differently (work by the authors to be reported).
3
-
THE ROlaE OF THE PHONONSTo assert a s we have done that the series in (18) is not t h e breather series of (17) a t all may seem presumptuous. However, we know from the TIM that the f r e e energy of sinh-G is /23/
(this checks with (18): put yo -r -yo bearing in mind there a r e no soliton solutions of sinh-G).
We also know /24/ that sinh-G has only phonons (and no kinks, antikinks or brehthers).
Thus the identification of the series in (18) a s a phonon series rather than a dressed form of the breather series of (17) is really that the series for sinh-G in (19) is derived from the sinh-G phonons; then the series in (18) follows by yo-' -yo reaching s-G.
The situation i s the more extraordinary in that a s integrable models both sinh-G and s-G a r e completely integrable /8,9/ with action-angle variables /8,9/. In s-G in particular the Hamiltonian H[pl in these variables is /8,9/
m - (20)
In this Hamiltonian M is the kink mass a s earlier and pi, p j a r e action variables (compare the expression (5) for the kink or antikink energies). Likewise 59 and 89 are action variables (compare (7)) and 89, 0 ( 89
<
Mr, i s the action variable for the internal degree of freedom of the breather.The final integral i s the phonons' contribution: w(k) is given by (16), the P(k) a r e action variables and they have canomical Q(k), 0 ( Q ( k )
<
277, angle variables such that t h e Poisson bracket {P(k), Q(kJ)}=
S(k-k') /8,9/. Now since sinh-G has neither kink, antikink or breather excitations its comparable H[p] must be(it is /8/). The only problem with this (apparently) i s that (21), found however by Fourier transform not t h e spectral transform /8,9/, is the H[p] for linear KG (9): i t has exactly the same form.
The situation is resolved in the following way /8,23/. Although sinh-G has no solitons and no breathers its thermodynamics is determined by large amplitude interacting phonons in thermal equilibrium: they interact through pair-wise phase shifts, and this statement applies to both the quantum sinh-G and the classical sinh-G /8,23,24/.
The case of s-G i s more bizarre: f i r s t in the quantum case there a r e no phonons /21,24/, only quantised kinks, antikinks and breathers; second in the classical case /21,22/ there a r e no breathers, but there a r e kinks, antikinks and classical phonons; third t h e classical limit of the quantum case of s-G /25/ shows that t h e quantum breathers become large amplitude classical phonons interacting through phase shifts exactly a s in t h e case of sinh-G described already.
The authors a r e slowly publishing the details of these (surely remarkable) features of the thermodynamics of the s-G model.
4
-
THE SINH-G MODEL I N A HEAT BATHAlthough the dynamics of a single soliton of t h e a-G travelling a t speed V and thermalisins to i t s equilibrium velocity in some time K ~ would be the ultimate aim of the work sketched in - ~
this paper, this calculation is not yet achieved. There is a substantial simplication if one uses the sinh-G model instead of the s-G (though there a r e now no kink o r antikink solutions).
The simplicity stems from (21) which shows sinh-G i s a bunch of (classical) oscillators. A bunch of quantum oscillators in a heat bath a t temperature T
=
T J - l>
0 satisfies the master equation /19/and [ak, akt] = S(k - k ' ) : y(k) a r e r a t e constants f o r each k.
Note t h a t (i) for bose oscillators this master equation (22) i s a solved eguation, eg. in equilibrium
( i i ) aktak o P(k) ( p a r t i c l e number) ;
(iii) a steady s t a t e solution i s p
=
e-rn / T r e-rn (which follows from R / ( 1+
R)=
e - m ( k ) ) . Here and in t h e master equation (22) H i s t h e Hamiltonian operator i n Schradinger representation: p i s t h e density matrix.
For a steady s t a t e solution, which i s all we can compute in t h i s paper, one should a s expected compute the partition function Z = Tr e-BH. But, with H a s given below (22),
The c l a s s i c a l l i m i t of t h i s is m
and t h i s diverges. However, if we use the cut-off 0
<
Ikl<
na-l,What h a s gone wrong? What has gone wrong i s t h a t we have failed t o a d d r e s s t h e problem of t h e thermodynamic limit.
5
-
TIIE THERMODYNAMIC LIMITWe must reach a finite density thermodynamic limit
-
achieved typically by using periodic boundary conditions of finite period L ( s a y ) s u c h t h a t for N particles i n L /8,21/l i m
N = ? i > O
L- L
For KG this has no special consequences (FL-I i s still given by FKG) b u t f o r (nonlinear) sinh-G the effect i s dramatic: we find
wh_ere_ Pn= O ( 1 ) : Pn <->P(Ti)dk (L+ -)and this implies P(E) is O(L): then L-IPn
=
L-t P(E)dg=
e ( k ) d k (say) and p(E) i s a finite densitiy. This explaiss t h e phrase, "large amplitude"
phonons used e a r l i e r : P(k) -t m. Note t h a t (27) y i e l d s w(k) p(k)dk a s L
I
+- .
d : -Y y0m2 [Xu(*')-k9w(k)]-I
(Ac i s t h e classical phonon;phonon phase shift and t h e "interaction" i s contained t h r o u g h (28) where the allowed modes k, a r e solutions of t h e system (28); kn r 2nn L-I, t h e f r e e field
Note that, for KG, yo
=
0 andkn =
k n in (28). One needs to know too t h a t (28) applies in the quantum case also /23/. But t h e n A, + Ab (see next) and P, in (28) has Pm=
0, 1, 2,...
a s forbose quantum oscillators.
There a r e actually two cases ( a t l e a s t ) /8,23/:
P
,
=
0, 1, 2,... for bosons with Ac + Ab a n d Pm=
0, 1 for ferrnions with Ac +Af.
One can choose e i t h e r case for t h e quantum theory and r e s u l t s a r e identical!We find /23/
with .Yo"
=
y o / ( l+
Yo/8n) and the smooth branch -2n (4
( 0 must be taken for t h e tan-1 in the fermion case. Then Ab(k, k t )=
Af(k, k')+
2n 8(kP-k) where 8 ( k )=
1, k>
0, 8 ( k )=
0, k<
0.
6
-
THE FREE ENERGYThe procedure now is t o calculate a n e n t r o p y S and s o t h e f r e e e n e r g y F: t h i s procedure substantially generalises the work of Yang and Yang on t h e repulsive NLS /8,26/
-
namely t o a boson description and t o t h e classical case also (we called i t method of 'generalised Bethe ansatz' /8,22,23/).A fundamental s e t of modes a r e t h e
G:
s e t k a function of t h eG
(ie. k, + k=
h ( g ) ) . /27/ilI, Then a s L + w (28) i s (now callB
+ k )J --m
The allowed modes d e f i n e a density of allowed s t a t e s f ( E ) ( o r f ( k ) ) (say) such t h a t
so (30) means (using bosons)
( l j ~ e f e r e n c e t o t h e Ref. /27/ shows t h e r e is_ a mistake i n t h e argument-of t h a t paper, where, a s a r e s u l t of a copying e r r o r , k, and kn a r e confused (roughly kn and kn a r e to be interchanged i n t h e 53 of /27/). A result i s o u r (32) and (44) below gain a wrong sign.
One a l s o has t h e energy E (Hamiltonian), momentum P, and number of p a r t i c l e s (bosons)
0 w
The e n t r o p y (for bosons) i s computed from t h e number of possible s t a t e s i n dk which i s [ L ( p
+
f ) dkl!/[L p dkl! [ L f dkl! (34)The e n t r o p y p e r unit length i s
s L
-'
=I
C(f+
p) l n ( f+
p) - f inf - p i n pldk -m(p is now the boson density not the density operator of (22)) and we minimise FL-I = ( E - B - l ~ ) ~ - l i e . we s e t S ( F L - ' ) / S ~ = 0. We readily find
m
--Q3 (35)
I f we define (f + p)p-' t exp *(k) we find the energies ~ ( k ) s a t i s f y
and we can go on t o show that
a
W e have thus regained (23) with however the condition that the energies ~ ( k ) satisfy (36)!
The classical limit of (37) with (36) puts In(1
-
R - E ( ~ ) ) + ln PE(k), and Ab +&
a s in (28), (this is the small yo "limit" of (29) in the boson form: there i s now a singularity a t k=
k', which not true of +).The iteration of this classical limit of (37) with (36) yields the asymptotic expansion of (19) exactly /23/. This explains how this result i s due to large amplitude phonons (thermodynamic limit L-lP, + p(k)dk, p(k) finite) interacting through phase shifts (the Ac).
One can either repeat the calculation for fermions instead of bosons (Pm
=
0, 1: Ac +4)
or transform (36) with (37)-
demonstrating strict equivalence. For the latter4 =
Ab-
2n 8-
while new energies Z(k) a r e defined by ln(1
+
e-@(k) ) = I l n ( 1-
e-@(k) ).
For the former it i s convenient in any case to introduce: a chemical potential LI by minimising the negative pressure -p rn FL-1-
LINL-~=
(E-
B'S-
p N ) ~ - l . The number of possible states (34) i s not changed but nowEither way t h e energies prove to safisfy (for finite u)
while
There a r e two interesting free particle limits: both arise where sin(%yow) + 0, but this is where yo/(l
+
yo/8v)=
0 o r 8n, so yo=
0 o r yo-. In both cases r ( k )=
~ ( k ) - p , just a e for a gas of f r e e bosons: since yO+O is t h e free KG, this i s expected but the result when Yo@ is not expected. In fact the repulsive NLS (2) (c>O) has the same integral equations like (39) with (40) in fermion form except that ~ ( k )=
k2 and Ap=
-2tan-l(c/(k-k')) (smooth branch) /26/.When t h e coupling constant c-, ~ ( k )
=
k2-u, a gas of free fermions. This is the impenetrable bose g a s /28/.7
-
THE QUANTUM MECHANICSFrom t h e results of 56 we we can obtain eigenenergies a t T=O (ie. t h e quantum mechanics).
We look for t h e aeros a t k= *kf of &(ak)=O. Then &uo i s fixed by this kf and E(k)
>
0, Ikl>
kf; z ( k )<
0, Ikl<
kf.Then from (39) as B1 = T + 0
I n analogy with t h e repulsive NLS /27,29/ energies El I E-Eo a r e excitation energies above the ground state energy Eo while
El =
E
(k*)- c
(kh). (43)El corresponds t o a fermion (in fermion description) in a state kp
>
kf together with a hole in a s t a t e k h<
kf. The energies t ( k p ) and E(kh) a r e solutions of (42).To g e t t h e ground s t a t e energy Eo, and 80 from (43) the excitation energy E
=
El+ %,
use~ ( k )
=
0, I k J>
kf and f ( k )=
density of (fermion) s t a t e s=
8 (k). Then ~ ( k ) becomes a solution of (32) in t h e form+K f
while
-kf
In this description p ( k ) , originally a density of bosons per unit length, must be interpreted a s a density of fermions per unit length. This is precisely what i s found by an a b initio fermion description (the e r r o r in reaching (44) from the boson description i s that something like a Bose-Einstein condensation occurs and the passage to (44) a t T
=
0 needs seperate analysis (this analysis i s not yet wholly done)).8
-
FURTHER AND FINAL REMARKSIn the case of the s-G i t is not yet clear how to couple quantum solitons and breathers (the only quantum excitations) to a heat bath a s was done for t h e phonons of sinh-G in 54. Note again that for the full quantum s-G there a r e only solitons (kinks and antikinks) and quantum breathers. Still the equilibrium analysis in terms of fermions can still be done: it yields a system of n-1 integral equations for distinct fermion energies: n
=
[8n yo-']=
integral p a r t , there a r e n-2 quantum breathers and one kink-antikink of double weight (see eg. Ref./25/ and references a s well a s work by t h e authors to be published).The quantum system has a semi-classical limit /25/: this can also be derived /22/ from an H[p]
Compared to (20) this neglects breathers: compared with (21) it will add kinks and antikinks (fermions) to phonons (bosons). It, nray be possiblc to write a quantum operator form for (46) which extends (22) to this case: if (22) is formally unchanged the damping of t h e solitons (kink and antikink) takes place only through interaction with t h e phonons which alone a r e coupled to the heat bath. The stability of the solitons perhaps makes such a model a good approximation.
Evidently a dynamical study of the thermalisation of s-G solitons will require much more work:
it is necessary to choose the phonon damping constants y ( k ) to model their thermalisation adequately. Then i t is necessary (in terms of the model just described) to calculate the decay time K ~ of a kink in terms of these ~ ( k ) . - ~
One of us (RKB) has already speculated /30/ on the calculation of Kk somewhat in the terms reported here. But it will be clear from reference to /30/ that we have advanced considerably in our understanding of the equilibrium steady state of both sinh-G and s-G, quantum or classical, since that paper.
Ref./30/ also compares with the work of Wada and Schrieffer /31/ who compute a diffusion constant D
=
0.516 woa2 (kgT/mOo2 wo2))' for the 0-four model(c
=
velocity) which though not integrable has for B>O long lived kinks and antikinks: Qo=
t ( ( A I / B ) ~ is a zero of the right side. By expanding Q about Qo there are "phonons" with dispersion wZ
=
21AIm-1+
cZkZ and wo=
( 2 1 ~ l m - ~ ) ) I . In D, kgT=
k1 while a is a lattice spacing much a s was used for ZKG earlier. To reach their result for D, the authors use finite amplitude phonons and work to second order: we here use pn for these amplitudes and following /31/, since<
pnz> -
kBT (= Pi) (<...> means thermal average),<
P,'>-
( k B ~ ) : to second order. Then D proves to be D-
(kBT)Z. Now the analysis to the free energy FL- of this paper also uses finite amplitude phonons a s explained, and the correction term to HL-I=
L-I Dd(kn) Pn introduced through (28) is of the order Pn2: since lpnI2
=
Pn, D - ( ~ B T ) ~ . However this result tells us only that there is a damping term perhaps like -Kka@/at to be included on the left side of the s-G equation (1). In Einstein's theory of Brownian motion for simple particles of mass mo and position x, D and K are related by mox -, mox+
m0Kx with<
xZ>=
2kBTt/~mo=
2Dt. More typically mo K=
kgT m o / ~ m o t where mot is an effective mass.This theory cannot apply directly if D - ( ~ B T ) ~ since then K increases a s T+O. Still the fluctuation dissipation theorem tells us that if -K &/at (= - K Q ~ ) adds to the left side of the s-G equation ( I ) , then we expect D
=
f ( T ) ~ - l where f(T) is a function of temperature to be determined. These rather imprecise considerations suggest again that (1) is to be replaced by@ti
+
K@t= qx -
m2 sin 6+
F(t)where K - ~ is an estimate of the time K ~ for a (biological) soliton of s-G type to reach - ~
equilibrium. We stress that all of the quantum and classical equilibrium results for FL-I found a s in 56 are exact /8,23/: they apply to any integrable model /8/ whose classical H[p]
takes the form (21) in action variables (with appropriate w(k) and A), and there are very many such systems. We stress too that the comparable results (eg. /21,22,32/ and work by the authors to be published) for integrable systems like s-G which classically have soliton solutions are similarly exact. We have exact results of the same sort for the classical Landau-Lifshitz model and the Toda lattice (Yu-zhong Chen, Ph.D thesis, U. of Manchester to be submitted). The results for the Toda lattice are relevant to the paper on solitons in DNA presented a t this meeting /33/.
In contrast with these exact equilibrium results we have still to provide an adequate analysis of the dynamical approach to equilibrium
-
though for sigh-G for example this is apparently provided by (22) with allowed modes k restricted to the k satisfying the quantum bose form of (28).Even so we must acknowledge that though the equilibrium theory is in good shape there is still much to be done before we can adequately- describe the dynamics of solitons on realistic models of biological systems.
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Additional note in proof: In t h e paragraph below that containing eqn. (18) we very tentatively suggest t h a t there might be a thermalisation process in which t h e undressed kink density (12) is actually realized if only transiently ( t h e final density i s certainly t h a t given in terms of free energy (18)). However, i t i s really p r e t t y clear that even if the biological molecule were initially subject only to Einstein type random force F ( t ) of Brownian motion due t o a surrounding solvent (say) then (12) i s realizable only if the temperature is v e r y low so that t h e r e a r e v e r y few phonons present. A s indicated below (18) solitons also d r e s s the solitons (they provide the multisoliton terms adding to (18)). We a r e not yet in a position t o say which dressing process dominates t h e thermalisation sequence.
