Tests of Nonlinear Resonse Theory. We compare the results of direct NEMD simulation against Kawasaki and TTCF for 2- particle colour conductivity.

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1 ess of Nonlinear Resonse heory We compare he resuls of direc NEMD simulaion agains Kawasaki and CF for 2- paricle colour conduciviy Direc CF, BK and RK J x DIR KAW CF KAW-BARE

2 Direc BK and RK DIR KAW CF KAW-BARE J x CF

3 Enropy 3 S() k d Γ f( Γ,)ln f( Γ,) B S () = k d [ ln f(,)] f(, ) B + Γ 1 Γ Γ = k d f(, ) B ΓΓ Γ Γ =+ kb d f(, ) Γ Γ Γ Γ = 3Nk α() B In he seady sae where averages of phase funcions are by definiion, ime independen, he enropy diverges (a a consan rae) owards -! In he absence of a hermosa he enropy of any Hamilonian sysem is a consan of he moion (Gibbs, 1902)!

4 Insabiliy of Phase Space rajecories 4 p Γ()+δΓ() δγ() ~ δγ() e λ Γ() Γ(0)+δΓ(0) Γ(0) q he equaions of moion for he infiniesimal angen vecors are, d d ΓΓ ( ( )) δγi() ( Γ) δγi() = δγi(), ( i= 1,.., 6N). Γ (31) In he infiniesimal limi, δγ i (0) 0, he formal soluion of his equaion can be wrien as, δγi() exp L[ ds( Γ()] s ) δγi( 0) L() δγi( 0), 0 (32)

5 he Lyapunov exponens are also he logarihms of he eigenvalues of he symmeric marix, Λ, Λ= lim( ) Λ( ) = lim( )[ L ( ) L( ) ] he Liouville equaion saes ha, (1/f)df/d = 3Nα. We can see ha he accessible volume of phase space, V~1/f, decreases o zero. ( ) 1/ 2 d df ( Γ, ) d ln V Γ() Γ = = λi = 3N α F = S / k e d d F e 6N i= 1 B (33) (34) 5 V(Γ(0)) V(Γ()) In he seady sae,

6 6 H = 0= P γv 2K α 0 xy 6N 2 λ = 3N α = ηγ ( ) γ i= 1 i 2 ηγ ( ) γ V= 3Nk α so, B V k B 6N ηγ ( )= kb 2 Vγ i= 1 λ ( γ) We his he Lyapunov Sum Rule for shear viscosiy. i (35) Symplecic Marices We define J, K, as, 0I I0 J K, I0, ;,, 0I, (36) where I is he 3N x 3N ideniy marix and 0 is he 3Nx3N null marix. For

7 Hamilonian sysems,, saisfies he infiniesimally symplecic condiion, 7 J = J (37) I is known ha his condiion is saisfied if he marix, can be wrien in he form, = A B C -A where he marices B and C are symmeric. L, saisfies he globally symplecic condiion if, Ideniies: (38) L JL J (39) If Lis g - symplecic : L J L J Using he fac ha J J= 1, L J L= J L J L J= 1 L J L J L = L -1-1 L L J L J L = L L J L J L = 1 L J L = J

8 If, Λ L L hen Λis g symplecic Λ J Λ= J = Λ J Λ Pf : ( L L) J L L= L L J L L = L J L = J If L is g-symplecic and L() 0 e L () sds 8 hen is i-symplecic. Differeniae wr, L () J L+ L J () L= 0 J= J Conversely if is i-symplecic hen L is g-symplecic. () J= J (), K hen a = 0 L ( 0) J L( 0) = J (Because L( 0) = 1) K #

9 9 Differeniae wr : d L () J L() = L () ( () J+ J ()) L() d If we use we see ha d L () J L() = 0, d Combining his wih #, we see ha L () J L() = J,. L is g - symplecic iff is i symplecic i-form of he Symplecic Eigenvalue heorem If J= J

10 u = ν u i i i J u = ν J u i i i J u = ν J u i i i and herefore if ν i is an eigenvalue of he i-symplecic marix, so oo is -ν i *. 10 g-form of he Symplecic Eigenvalue heorem If, L is g - symplecic and L x = λx ( J x) L= x J L Bu from he g - symp condiion, L J L J J L= L J So, 1 1 ( J x) L= x L J

11 From he eigenvalue equaion ( L x) = x L = λx and x 1 1 L = λ x Subsiuing gives, 1 1 ( J x) L= λ x J = λ ( J x) and ( J x) is an eigenvecor of Lwih eigenvalue J xis an eigenvecor of L wih eigenvalue λ λ -1 is an eigenvalue of L -1 λ -1 11

12 Conjugae Pairing Rule for Lyapunov Exponens 12 he Lyapunov exponens are he logarihms of he eigenvalues of he real symmeric / marix Λ( ) = L () L() 1 2, and Λ lim Λ( ). If L is g-symplecic hen Λ is g- [ ] = symplecic so ha, Λ J Λ= Λ J Λ= J Clearly herefore, J Λ= Λ 1 J If u i is an eigenvecor of Λ wih real eigenvalue ν i, Λ u = ν u i i i J Λ u = ν J u 1 i i i Λ J ui = νij ui Rearranging gives ha, Λ J u = ν 1 J u i i i his implies ha J u i is an eigenvecor of Λ wih eigenvalue ν i 1. he Lyapunov exponen corresponding o he eigenvecor u i is λ i = ln(ν i ), and ha corresponding o

13 J u i is λ i* = ln(ν i ). hus for dynamical sysems wih an i-symplecic local sabiliy marix (or equivalenly wih a g-symplecic angen propagaor marix L), Lyapunov exponens occur in conjugae pairs which sum o zero. 13 hermosaed Hamilonian sysems. L( ) = exp ds( s) = exp ds[ ' ( s) α( s) 1/ 2] L' ( ) e L 0 L 0 α/ 2 where α 1 α() sds 0. ' has he i-symplecic symmery so ' is i-symplecic and L' is g-symplecic. Leing Λ' = lim L' () L' () [ ] 1/ 2 we know ha Λ' is also g-symplecic and since, Λ=Λ'.exp[ α/2] and, Λ J Λ = J, so α α Λ J Λ= e Λ' J Λ' = e J J Λ= e α Λ 1 J. If

14 Λ u = ν u i i i 14 α J Λ u = e Λ 1 () J u = ν J u i i i i 1 α Λ() J ui = νi e J ui hus if u i is an eigenvecor wih eigenvalue ν i, J.u i is an eigenvecor wih eigenvalue ν 1 i e α. he Lyapunov exponen corresponding o he eigenvecor u i is λ i = ln(ν i ), and ha corresponding o J u i is λ i* = ln(ν i ) α. his in urn implies ha if λ i is a Lyapunov exponen, is conjugae exponen is λ i* = λ i α and he sum of he conjugae pairs of exponens is α, independen of he pair index. his is known as he Conjugae Pairing Rule. LF ( ) = e nk( λ ( Fe) + λ N( Fe)) 2 F e