Simple model for one-dimensional crystal in solid-state physics
TheToda lattice , introduced byMorikazu Toda (1967 ), is a simple model for a one-dimensional crystal insolid state physics . It is famous because it is one of the earliest examples of a non-linearcompletely integrable system .
It is given by a chain of particles with nearest neighbor interaction, described by the Hamiltonian
H ( p , q ) = ∑ n ∈ Z ( p ( n , t ) 2 2 + V ( q ( n + 1 , t ) − q ( n , t ) ) ) {\displaystyle {\begin{aligned}H(p,q)&=\sum _{n\in \mathbb {Z} }\left({\frac {p(n,t)^{2}}{2}}+V(q(n+1,t)-q(n,t))\right)\end{aligned}}} and the equations of motion
d d t p ( n , t ) = − ∂ H ( p , q ) ∂ q ( n , t ) = e − ( q ( n , t ) − q ( n − 1 , t ) ) − e − ( q ( n + 1 , t ) − q ( n , t ) ) , d d t q ( n , t ) = ∂ H ( p , q ) ∂ p ( n , t ) = p ( n , t ) , {\displaystyle {\begin{aligned}{\frac {d}{dt}}p(n,t)&=-{\frac {\partial H(p,q)}{\partial q(n,t)}}=e^{-(q(n,t)-q(n-1,t))}-e^{-(q(n+1,t)-q(n,t))},\\{\frac {d}{dt}}q(n,t)&={\frac {\partial H(p,q)}{\partial p(n,t)}}=p(n,t),\end{aligned}}} whereq ( n , t ) {\displaystyle q(n,t)} n {\displaystyle n}
andp ( n , t ) {\displaystyle p(n,t)} m = 1 {\displaystyle m=1}
and the Toda potentialV ( r ) = e − r + r − 1 {\displaystyle V(r)=e^{-r}+r-1}
Soliton solutions are solitary waves spreading in time with no change to their shape and size and interacting with each other in a particle-like way. The general N-soliton solution of the equation is
q N ( n , t ) = q + + log det ( I + C N ( n , t ) ) det ( I + C N ( n + 1 , t ) ) , {\displaystyle {\begin{aligned}q_{N}(n,t)=q_{+}+\log {\frac {\det(\mathbb {I} +C_{N}(n,t))}{\det(\mathbb {I} +C_{N}(n+1,t))}},\end{aligned}}} where
C N ( n , t ) = ( γ i ( n , t ) γ j ( n , t ) 1 − e κ i + κ j ) 1 < i , j < N , {\displaystyle C_{N}(n,t)={\Bigg (}{\frac {\sqrt {\gamma _{i}(n,t)\gamma _{j}(n,t)}}{1-e^{\kappa _{i}+\kappa _{j}}}}{\Bigg )}_{1<i,j<N},} with
γ j ( n , t ) = γ j e − 2 κ j n − 2 σ j sinh ( κ j ) t {\displaystyle \gamma _{j}(n,t)=\gamma _{j}\,e^{-2\kappa _{j}n-2\sigma _{j}\sinh(\kappa _{j})t}} whereκ j , γ j > 0 {\displaystyle \kappa _{j},\gamma _{j}>0} σ j ∈ { ± 1 } {\displaystyle \sigma _{j}\in \{\pm 1\}}
The Toda lattice is a prototypical example of acompletely integrable system . To see this one usesFlaschka 's variables
a ( n , t ) = 1 2 e − ( q ( n + 1 , t ) − q ( n , t ) ) / 2 , b ( n , t ) = − 1 2 p ( n , t ) {\displaystyle a(n,t)={\frac {1}{2}}{\rm {e}}^{-(q(n+1,t)-q(n,t))/2},\qquad b(n,t)=-{\frac {1}{2}}p(n,t)} such that the Toda lattice reads
a ˙ ( n , t ) = a ( n , t ) ( b ( n + 1 , t ) − b ( n , t ) ) , b ˙ ( n , t ) = 2 ( a ( n , t ) 2 − a ( n − 1 , t ) 2 ) . {\displaystyle {\begin{aligned}{\dot {a}}(n,t)&=a(n,t){\Big (}b(n+1,t)-b(n,t){\Big )},\\{\dot {b}}(n,t)&=2{\Big (}a(n,t)^{2}-a(n-1,t)^{2}{\Big )}.\end{aligned}}} To show that the system is completely integrable, it suffices to find aLax pair , that is, two operatorsL(t) andP(t) in theHilbert space of square summable sequencesℓ 2 ( Z ) {\displaystyle \ell ^{2}(\mathbb {Z} )}
d d t L ( t ) = [ P ( t ) , L ( t ) ] {\displaystyle {\frac {d}{dt}}L(t)=[P(t),L(t)]} (where [L , P ] = LP - PL is theLie commutator of the two operators) is equivalent to the time derivative of Flaschka's variables. The choice
L ( t ) f ( n ) = a ( n , t ) f ( n + 1 ) + a ( n − 1 , t ) f ( n − 1 ) + b ( n , t ) f ( n ) , P ( t ) f ( n ) = a ( n , t ) f ( n + 1 ) − a ( n − 1 , t ) f ( n − 1 ) . {\displaystyle {\begin{aligned}L(t)f(n)&=a(n,t)f(n+1)+a(n-1,t)f(n-1)+b(n,t)f(n),\\P(t)f(n)&=a(n,t)f(n+1)-a(n-1,t)f(n-1).\end{aligned}}} wheref(n+1) andf(n-1) are the shift operators, implies that the operatorsL(t) for differentt are unitarily equivalent.
The matrixL ( t ) {\displaystyle L(t)} inverse scattering transform for theJacobi operator L . The main result implies that arbitrary (sufficiently fast) decaying initial conditions asymptotically for larget split into a sum of solitons and a decayingdispersive part.
Krüger, Helge; Teschl, Gerald (2009), "Long-time asymptotics of the Toda lattice for decaying initial data revisited",Rev. Math. Phys. ,21 (1):61– 109,arXiv :0804.4693 Bibcode :2009RvMaP..21...61K ,doi :10.1142/S0129055X0900358X ,MR 2493113 ,S2CID 14214460 Teschl, Gerald (2000),Jacobi Operators and Completely Integrable Nonlinear Lattices ISBN 978-0-8218-1940-1 MR 1711536 Teschl, Gerald (2001),"Almost everything you always wanted to know about the Toda equation" ,Jahresbericht der Deutschen Mathematiker-Vereinigung ,103 (4):149– 162,MR 1879178 Eugene Gutkin, Integrable Hamiltonians with Exponential Potential, Physica 16D (1985) 398-404.doi :10.1016/0167-2789(85)90017-X Toda, Morikazu (1967), "Vibration of a chain with a non-linear interaction",J. Phys. Soc. Jpn. ,22 (2):431– 436,Bibcode :1967JPSJ...22..431T ,doi :10.1143/JPSJ.22.431 Toda, Morikazu (1989),Theory of Nonlinear Lattices , Springer Series in Solid-State Sciences, vol. 20 (2 ed.), Berlin: Springer,doi :10.1007/978-3-642-83219-2 ,ISBN 978-0-387-10224-5 MR 0971987
