Chapter 2 Dynamics

Chapter 2

Dynamics

We study here the dynamics of the infinite chain and we set the problem of itsergodicity. We need first to give a proper definition of ergodicity for an infinitedynamics. We need some stochastic perturbation in order to prove something aboutthis ergodicity. We introduce then some stochastic perturbation acting only on thevelocities such that will conserve the total energy and momentum of the chain.

2.1 Dynamics of the infinite system

In order to avoid technical di!culties we assume the potential V satisfies

V !(r)2 ! CV (r), V !!(r) ! C (2.1.1)

Basically we require that V grows to infinity quadratically.

Let T > 0 be fixed. We consider now the system in the infinite lattice Z. Asbefore, rj = qj " qj"1 is the interparticle distance. Let us denote " = (R2)Z theconfiguration space and ! = (pi, ri)i#Z # " the generic configuration. We introducethe space D(") composed of smooth local functions having continuous and boundedpartial derivatives at every order.

The hamiltonian dynamics is given by the solution of the infinite system ofdi#erential equations

rj(t) = pj(t)" pj"1(t)

pj(t) = V !(rj+1(t))" V !(rj(t)), j # Z.(2.1.2)

and the formal generator of the dynamics is the Liouville operator

A =!

j#Z

"(pj " pj"1) "rj + (V !(rj+1)" V !(rj)) "pj

#(2.1.3)

9

10 CHAPTER 2. DYNAMICS

with domain D(").

The existence of the solution for (2.1.2) can be proven for a wide class ofinitial conditions, in particular for a set of configurations that has measure one forany Gibbs grand-canonical distribution

dµgc!,p," =

$

j#Z

e""(Ej"ppj"!rj)

Z(#$, #p, #)drjdpj . (2.1.4)

for all parameters $, p, # > 0. We will prove this existence by an iteration schemeand we will show the dynamics is approximated by finite dimensional dynamics, seesection 2.2. Notice that the momentum is formally conserved, which follows fromthe translation invariance of the dynamics. This is why we have the third parameterp in the stationary measures.

A set of initial conditions can be defined in the following way. For a > 0, wedenote

"a =

%! = (rj, pj)j#Z # (R2)Z : $!$2

a =!

j#Z(p2

j + r2j )e

"a|j| < %&

equipped with the topology induced by the norm $ · $a. It is easy to check that anyprobability measure % such that

'(r2

j + p2j)d% ! Cebj with b < a gives measure one

to "a. In particular any Gibbs measure µgc!,p," satisfies this condition since

(r2e""(V (r)"!r)dr ! C [Z($# + 1, #) + Z($# " 1, #)]

Lemma 2.1.1 For any ! # "a, there exists a unique continuous process !(t) =(ri(t), pi(t))i#Z belonging to "a and satisfying (2.1.2) with initial condition !(0) = !.

Proof We define recursively !(n)(t) = (r(n)j (t), p(n)

j (t))j#Z by

dr(n+1)j (t) = (p(n)

j (t)" p(n)j"1(t))dt

dp(n+1)j (t) = (V !(r(n)

j+1(t))" V !(r(n)j (t)))dt, j # Z.

and !(0) = !. By Cauchy-Schwarz’s inequality, there exists a constant C > 0 suchthat for any t # [0, T ],

$!(n+1)(t)" !(n)(t)$2a ! CT

( t

0

ds$!(n)(s)" !(n"1)(s)$2a (2.1.5)

2.1. DYNAMICS OF THE INFINITE SYSTEM 11

By induction, it follows that for t # [0, T ]

$!(n+1)(t)" !(n)(t)$2a !

T (CT )n

n!

( T

0

dt $!(1)(t)" !(0)(t)$2a

! 2(CT 4 + T 2)(CT )n

n!$!$2

a

If T is su!ciently small then)

n(CT )n/n! < %. It follows that !(n)(t) convergesto some !(t) in ("a, $ ·$ a). Moreover, the convergence is uniform on [0, T ] and !(t)is solution of (2.1.2). It is easy to extend the domain of existence for arbitrary largeT . Uniqueness (in "a) is a simple consequence of estimate (2.1.5). !

For any ! # "a, the quantity*)

j e"a|j|Ej(!)+

is finite (thanks to (2.1.1)).

Starting from !, (2.1.1) gives

d

dt

,!

j#Ze"a|j|Ej(!(t))

-! C

!

j#Ze"a|j|Ej(!(t)) (2.1.6)

and then the following a priori bound

.!

j#Ze"a|j|Ej(!(t))

/! C0e

c1t

.!

j#Ze"a|j|Ej(!)

/(2.1.7)

By this way we define a semigroup (Pt)t$0 on the Banach space C("a) of localbounded continuous functions on "a:

&f # C("a), &! # "a, (Ptf)(!) = f(!t)

where !t = {rj(t), pj(t); j # Z} is the solution of (2.1.2). Moreover (Pt)t is contrac-tive w.r.t. the L2-norm associated to µgc

!,p,". It follows that Pt can be extended to asemi-group of contraction on L2(", µgc

!,p,").

The proof given in lemma 2.4.2 is easily adapted in the deterministic contextand gives the following lemma

Lemma 2.1.2 There exists a closed extension of the Liouville operator A (see(2.1.3)) in L2(", µgc

!,p,") such that the space D(") is a core. This closed extension isthe generator of the strongly continuous semigroup (Pt)t defined above. In particularfor all values of $, p, # > 0, the Gibbs measures µgc

!,p," are stationary.


2.2 Finite-dimensional approximation

The existence of the infinite dynamics can also be proven by approximation by finitedynamics. We outline the arguments since they are given with all the details for thestochastic dynamics.

We can define, for any n, the finite dimensional dynamics generated by

An =n!

j="n

"pj("rj " "rj+1) + (V !(rj+1)" V !(rj)) "pj

#(2.2.1)

Observe that this dynamics changes only p"n, . . . , pn and r"n, . . . , rn+1 keeping)n+1j="n rj constant. All the other (infinite) variables remain constant, determined

by the initial conditions. Notice that all grand canonical Gibbs measures {µgc!,0,", $ #

R, # > 0} are stationary for these dynamics, for all n.

The operator An is the generator of !n(t) = {rni (t), pn

i (t), i # Z} solution ofthe evolution equations starting from the configuration !, i.e.

r(n)i (t) = p(n)

i (t)" p(n)i"1(t) i = "n, . . . , n + 1

p(n)i (t) = V !(r(n)

i+1(t))" V !(r(n)i (t)), i = "n, . . . , n.

(2.2.2)

with the coordinates (qi(t), pi(t)) = (qi(0), pi(0)) if i '# {"n, . . . , n}.Let P t

n the corresponding semigroup generated by this evolution:

0P t

n&1(!) = &(!n(t)).

We need later on some uniform control in the finite dimensional approximationof the infinite dynamics. The arguments given in the proof of lemma 2.4.3 can beeasily adapted to prove

Lemma 2.2.1 Let a > 0 and ! # "a, then

• For any i # Z (r(n)i (t), p(n)

i (t)) converges, uniformly in [0, T ], to (ri(t), pi(t))as n goes to infinity .

• For any bounded Lipschitz local function &, there exist constants Cn = Cn(T,&) (0 as n (% such that:

supt#[0,T ]

22P tn&(!)" P t&(!)

22 ! Cn

,!

j#Ze"a|j|Ej(!)

-

2.3. ERGODICITY 13

2.3 Ergodicity

Ergodicity is one of the main open problem for Hamiltonian systems, in fact wethink there is not a general agreement on what it means for an infinite system.

For us ergodicity will mean a characterization of the stationary translationinvariant probability measure, in a class of locally regular measure, as convex com-bination of Gibbs measures.

Definition 2.3.1 We say that the dynamics defined by (2.1.2) is ergodic if anyprobability measure % on the configuration space that

1. has finite density entropy : )C > 0,&$ " Z, H!(%|µgc0,0,1) ! C|$|

2. is translation invariant,

3. is stationary, i.e. for any function F (r, p) # D(")(AF d% = 0 (2.3.1)

is a convex combination of Gibbs measures µgc!,p,".

Remark 2.3.2 In the first assumption µgc0,0,1 does not play any role and can be

replaced by any Gibbs measure µgc!,p,".

It is clear that this property is not always true. The easier example is theharmonic case, i.e. V quadratic. We are tempted to conjecture that for genericnon-linear dynamics the system is ergodic. But for Toda lattice interaction V (r) =aer " r " b, the dynamics is completely integrable in its finite dimensional version(like the harmonic case) and constitute another conterexample. So it is not clear onwhich class of anharmonic V the ergodic property can be conjectured.

The idea is that the nonlinearity should mess up su!ciently the distributionof the velocity.

Theorem 2.3.3 Let % satisfy the three conditions of definition 2.3.1, and further-more the distribution of the velocities conditioned to the position %(dp|r) is ex-cheangeable. Then % is a convex combination of Gibbs measures µgc

!,p,".

Let Finv the '-field of the sets of " invariant for translations. The first inproving Theorem 2.3.3 is to show the conditional independence

%(dr, dp|Finv) = %(dr|Finv)%(dp|Finv) (2.3.2)


Proof of (2.3.2) Let &(r) and ((p) two local continuous function respectivelyof the r’s and the p’s. Then by excheangeability of %(dp|r), if we define

)k(r) =

((($kp)%(dp|r)

we have )k(r) = )0(r). On the other hand by the translations invariance of % wehave

()k(r)&(r)d%(r) =

((($kp)&(r)d% =

(((p)&($"kr)d% =

()0($kr)&(r)d%(r)

for any arbitrary function &, that imply )0(r) = )0($kr). Form this is easy to obtain(2.3.2). !

Because % has finite entropy density the entropy inequality (proposition A.1.1)gives the following bound on the energy density

)C > 0,&j # Z,

(Ejd% ! C (2.3.3)

Then the ergodic theorem allows us to define % a.s. the following quantities

z0 = lim#%&

z#0 = lim

#%&

1

*

#!

j=1

rj = %(rj|Finv)

z1 = lim#%&

z#1 = lim

#%&

1

*

#!

j=1

pj = %(pj|Finv)

z2 = lim#%&

z#2 = lim

#%&

1

*

#!

j=1

p2j = %(p2

j |Finv)

and we denote by z(r, p) = (z0, z1, z2) the corresponding random vector (with valueson R*R*R+). We also define the Finv-measurable random variable # = (z2"z2

1)"1.

Observe that # can be defined also by

#"1 = %(pj(pj " z1)|Finv) (2.3.4)

Lemma 2.3.4 For any function F (r, p) # D(") and any function h in D(R*R*R+) we have (

h(z(r, p))AF (r, p) d%(r, p) = 0 .

2.3. ERGODICITY 15

Proof : Because of the stationarity of % we have

0 =

(A[h(z#(r, p))F (r, p)] d%(r, p)

=

(h(z#(r, p))AF (r, p) d%(r, p) +

([Ah](z#(r, p))F (r, p) d%(r, p)

so we only need to show that

lim#%&

([Ah](z#(r, p))F (r, p) d%(r, p) = 0

Since F has compact support and +h is bounded, we only need to prove

lim#%&

( 22Az#$(r, p)

22 d%(r, p) = 0, + = 0, 1, 2. (2.3.5)

This is easy for + = 0, 1. Indeed

Az#0(r, p) =

p# " p0

*, Az#

1(r, p) =V !(r#+1)" V !(r1)

*

and since'

(|pl|+ |V !(r#)|)d% ! C, C independent of *, (2.3.5) follows immediatelyfor + = 0, 1.

For + = 2 we have

Az#2(r, p) =

1

*

#!

j=1

pj(V!(rj+1)" V !(rj))

=1

*

#!

j=1

(pj " z1)(V!(rj+1)" V !(rj)) + z1

V !(r#+1)" V !(r1)

*

About the second term, we can bound it using Schwarz inequality

1

*

(|z1(V

!(r#+1)" V !(r1))| d%(r, p) ! 1

*

( 0z21/2 + V !(r0)

21d%(r, p)

that goes to 0 as * (%. Taking the expectation of the square of the first term wehave

( 3

4( ,

1

*

#!

j=1

(pj " z1)(V!(rj+1)" V !(rj))

-2

%(dp|r)

5

61/2

d%

=1

*

( * #!

i,j=1

7((pj " z1)(pi " z1) %(dp|r)

8*

* (V !(rj+1)" V !(rj))(V!(ri+1)" V !(ri))

+1/2

d%(r)

(2.3.6)


Since %(dp|r) is excheangeable, by the Hewitt-Savage theorem (ref..) it is aconvex combination of product probability measures, i.e.

%(dp|r) =

(d%(,, r)

$

i#Zf(dpi, ,) . (2.3.7)

and z1(r, p) = z1(p), we have(

(pj " z1)(pi " z1)%(dp|r) =

(d%(,, r)

((pj " p)(pi " p)

$

k#Zf(dpk, ,)

= -i,j

(d%(,, r)

((p0 " p)2

$

k#Zf(dpk, ,) = -i,j#

"1(r)

Consequently (2.3.6) is equal to

1

*

(#"1/2(r)

9::;#!

i=1

(V !(ri+1)" V !(ri))2 %(dr)

that, again by Schwarz inequality, is bounded by

1

2,*

(#"1(r) d%(r) +

,

2*

#!

i=1

((V !(ri+1)" V !(ri))

2 d%(r) (2.3.8)

where , is an arbitrary positive number. Recall that there exists a constant C > 0such that

(V !)2 ! CV (2.3.9)

Hence we have (2.3.8) is bounded above by

1

2,*

(#"1(r) d%(r) +

2C,

*

#+1!

i=1

(V (ri) d%(r) (2.3.10)

We recall that #"1 is in L1(%) and'Ejd% ! K. Then we have

lim sup#%&

1

*

(#"1/2(r)

9::;#!

i=1

(V !(ri+1)" V !(ri))2 %(dr)

! lim sup%%0

lim sup#%&

%1

2,*

(#"1(r) d%(r) +

2C,

*

#+1!

i=1

(V (ri) d%(r)

&

! lim sup%%0

lim sup#%&

<1

2,*

(#"1(r) d%(r) + 4CK,

=

= 0

2.3. ERGODICITY 17

!

Proof of Theorem 2.3.3.

It follows from Lemma 2.3.4 that for every F (r, p) # D(")(AF (r, p) d%(r, p|z) = 0 . (2.3.11)

where d%(r, p|z) is the measure % conditioned to the values of z.

Choosing in (2.3.11) the function

F = (pj " z1)&(r)

with &(r) # D(RZ) that we specify later, we obtain

0 =

( %!

i

(pj " z1)pi

0"ri&" "ri+1&

1+ (V !(rj+1)" V !(rj)) &(r)

&d%(r, p|z)

=

( %!

i

7((pj " z1)pi%(dp|r,Finv)

8 0"ri&" "ri+1&

1+ (V !(rj+1)" V !(rj)) &(r)

&d%(r|z)

=

( %!

i

7((pj " z1)pi%(dp|Finv)

8 0"ri&" "ri+1&

1+ (V !(rj+1)" V !(rj)) &(r)

&d%(r|z)

=

( >#"1

0"rj&" "rj+1&

1+ (V !(rj+1)" V !(rj)) &(r)

?d%(r|z)

(2.3.12)

where we have used the fact that z is Finv mesurable and (2.3.2).

Define(#(r) = e

P!i=1("V (ri)"&ri)

where . = .(z0, #). Notice that . and # are just function of z, so we can treat themas constant under %(dr|z). Choosing

&(r) = )(r)(#(r),

with )(r) a local smooth function, we get for any j = 1, . . . , *" 1:(

#"10"rj)" "rj+1)

1(l(r) d%(r|z) = 0 .

We choose now

)(r) = )b(r)g

,#!

i=1

ri

-)0(r1, . . . , r#"1),


where )b is a local function not depending on r1, . . . , r#, and g is a smooth functionon R. Since

"rj)(r)" "rj+1)(r) = )b(r)g

,#!

i=1

ri

-0"rj)0(r)" "rj+1)0(r)

1

if j = 1, . . . , * " 1, we can further condition on)#

k=1 rk = *r and on the exteriorr for the averageis a bad notation configuration {ri, i '= 1, . . . , *}, and obtain, for all j = 1, . . . , *" 1,

( 0"rj)0(r)""rj+1)0(r)

1(l(r)

%

,dr1, . . . , dr#

222#!

k=1

rk = *r, ri, i '= 1, . . . , *, z

-= 0

(2.3.13)

This is enough to characterize the measure

(l(r)%

,dr1, . . . , dr#

222#!

k=1

rk = *r, ri, i '= 1, . . . , *, z

-=

= e"#&r#$

i=1

e"V (ri)%

,dr1, . . . , dr#

222#!

k=1

rk = *r, ri, i '= 1, . . . , *, z

-

as the Lebesgue measure on the hyperplane {(r1, . . . , r#) :)#

k=1 rk = *r} (up to amultiplicative constant). In particular it follows that

%(dr1, . . . , dr#|ri, i '= 1, . . . , *, z) =e"

P!i=1("V (ri)"&ri)

Z(., #)#dr1 . . . dr#.

which implies

%(dr|z) =$

j#Z

e"("V (ri)"&ri)

Z(., #)(2.3.14)

and also that

%(dr|Finv) =$

j#Z

e"("V (ri)"&ri)

Z(., #)(2.3.15)

Similarly, choosing in (2.3.11) the function

F = ((p)(rj " z0)

2.4. CONSERVATIVE STOCHASTIC DYNAMICS 19

we have

0 =

( %(pj " pj"1)((p) +

!

i

(V !(ri+1)" V !(ri)) (rj " z0)"pi((p)

&d%(r, p|z)

(2.3.16)Since

(V !(ri)(rj " z0)"pi((p) d%(r, p|z)

=

(d%Finv(·|z)

7(V !(ri)(rj " z0)%(dr|Finv)

8 7("pi((p)%(dp|Finv)

8

= -i,j#"1

("pi((p) d%(p|z)

and (2.3.16) became

( "(pj " pj"1)((p) + #"1("pj " "pj)((p)

#d%(p|z) = 0 (2.3.17)

which is enough to characterise d%(p|z) as

%(dp|z) =$

j#Z

e""(pi"z1)2/2

@2/#"1

= %(dp|Finv) (2.3.18)

and finally we have, by (2.3.2)

%(dr, dp|z) = %(dr|z)%(dp|z) = µgcz (2.3.19)

!

2.4 Conservative stochastic dynamics

The proof exposed in section 2.3 is based on the assumption that the stationarydistribution of the velocity, conditioned on the positions, are convex combination ofgaussians. Nowhere we used the non-linearity of the interaction, that in a hamilto-nian deterministic dynamics should be the cause of such mixing of velocities. Thisis a very di!cult problem, that we do not even know how to attack mathematically.We choose an easier path here which is to approximate the mixing e#ect due the


non-linearities of the dynamics by adding some stochastic terms to it. The por-pouse of this stochastic term is to create this mixing of the velocities so that we canapply the argument of section 2.3. We also would like that the total momentumand total energy is conserved by this stochastic mechanism, and that has a localnature. A simple way is to exchange, in a continuous random way, the momentumof each three consecutive particles pj"1, pj, pj+1, in such way that pj"1+pj +pj+1 andp2

j"1 + p2j + p2

j+1 are both conserved. In the corresponding R3 configuration space ofthese momentum, it is left a circle where the noise can act. Define the vector fields

Yi,j,k = (pk " pj)"pi + (pi " pk)"pj + (pj " pi)"pk. (2.4.1)

and for consecutive particlesYj = Yj"1,j,j+1

We define now the second order operator

S =!

i

Y 2i (2.4.2)

In its finite dimensional version (i.e. Sn =)n"1

j=2 Y 2j on Rn) it is an hypoelliptic

operator, in the sense that the Lie algebra generated by {Y2, . . . , Yn"1} is the fulltangent space to the (n" 2)-dimensional sphere

Sn,p,K =

%(p1, . . . , pn) # Rn :

n!

j=1

pj = np,n!

j=1

p2j = nK

&.

This is enough to characterize all stationary measure for the stochastic dynamicsgenerated by Sn as the uniform measures on the corresponding sphere Sn,p,K .

The problem is much more complicate for the infinite dynamics. This is definedas the Markov process on " = (R2)Z generated by the formal generator

L = A+ 0S. (2.4.3)

The ode describing the evolution are now substiuted by the following stochasticdi#erential equations:

drj(t) = (pj(t)" pj"1(t))dt

dpj(t) = (V !(rj+1(t))" V !(rj(t)))dt +0

6%(4pj + pj"1 + pj+1)dt

+

A0

3

!

k="1,0,1

(Yj+kpj) dwj+k(t), j # Z.

(2.4.4)


Here {wj(t)}j#Z are independent standard Wiener processes and % is the discretelaplacian on Z,

%f(z) = f(z + 1) + f(z " 1)" 2f(z).

The existence of the infinite dynamics is done similarly to the deterministiccase. To get the approximation by finite dimensional dynamics one defines a dynam-ics generated by Ln = An + 0Sn, then we prove that this converges to the infinitedynamics in a set of initial conditions that has measure one for any Gibbs measure.Notice that also here all Gibbs measure with null momentum average are stationaryfor these finite dynamics.

For the set of initial conditions we choose as in the deterministic case the set

"a =

%! = (rj, pj)j#Z # (R2)Z : $!$2

a =!

j#Z(p2

j + r2j )e

"a|j| < %&

, a > 0

which has measure one with respect to any Gibbs measure.

Lemma 2.4.1 For any ! # "a, there exists a unique stochastic process !(t) =(ri(t), pi(t))i#Z belonging to "a and satisfying (2.4.4) with initial condition !(0) = !.

Proof We define recursively !(n)(t) = (r(n)i (t), p(n)

i (t))i#Z by

dr(n+1)j (t) = (p(n)

j (t)" p(n)j"1(t))dt

dp(n+1)j (t) = (V !(r(n)

j+1(t))" V !(r(n)j (t)))dt +

0

6%(4p(n)

j + p(n)j"1 + p(n)

j+1)dt

+

A0

3

!

k="1,0,1

BYj+kp

(n)j

Cdwj+k(t), j # Z.

and !(0) = !. By (2.1.1), there exists a constant C > 0 such that

E'

D$!(n+1)(t)" !(n)(t)$2

a

E! C

( t

0

dsE'

D$!(n)(s)" !(n"1)(s)$2

a

E

By induction, it follows that for t # [0, T ]

E'

D$!(n+1)(t)" !(n)(t)$2

a

E! (CT )n/n!

If T is su!ciently small then)

n(CT )n/n! < % and !(n)(t) converges to !(t) # "a

in L2("a, P'). The uniqueness (in "a) is standard. !


Starting from an initial configuration ! # "a, conservation of the energy bythe dynamics gives the following a priori bound

E'

.!

j

e"a|j|Ej(!(t))

/! C0e

c1t

.!

j

e"a|j|Ej(!)

/(2.4.5)

By this way we define a semigroup (Pt)t$0 on the Banach space C("a) of localbounded continuous functions on "a. For any f # C("a), we have

&! # "a, (Ptf)(!) = E' [f(!t)]

where !t = {rj(t), pj(t); j # Z} is the solution of (2.4.4). Moreover (Pt)t is contrac-tive w.r.t. the L2-norm associated to µgc

!,p,". It follows that Pt can be extended to asemi-group of contraction on L2(", µgc

!,p,").

By continuity of the paths !t and the bounded convergence theorem, we havethat for any & # D("),

limt%0

µgc!,p,"((Pt&" &)2) = 0

Since any function in L2(", µgc!,p,") can be approximated by a sequence of elements

of D(") and Pt is contractive, it follows that Pt is a strongly continuous semigroupof contractions on L2(", µgc

!,p,").

By Ito’s formula, we have that for any & # D(")

&! # "0, (Pt&)(!) = (P0&)(!) +

( t

0

(PsL&)(!)ds

This shows that any & # D(") belongs to the domain of the generator L of theL2-semigroup (Pt)t$0 and that L and L coincide on D("). It follows that D(") is acore for L. we have proved the following lemma

Lemma 2.4.2 There exists a closed extension of L (see (2.4.3) ) in L2(", µgc!,0,")

such that the space D(") is a core. This closed extension is the generator of thestrongly continuous semigroup (Pt)t defined above.

We now prove that the infinite volume dynamics is well approximated by thefinite dimensional dynamics !n(t) = {rn

i (t), pni (t), i # Z}. It is defined by the

following stochastic di#erential equations

r(n)i (t) = p(n)

i (t)" p(n)i"1(t) i = "n, . . . , n + 1

p(n)i (t) = V !(r(n)

i+1(t))" V !(r(n)i (t)) +

0

6%(4p(n)

i + p(n)i"1 + p(n)

i+1)dt

+

A0

3

!

k="1,0,1

BYi+kp

(n)i

Cdwi+k(t), i = "n, . . . , n.

(2.4.6)


with the coordinates (qi(t), pi(t)) = (qi(0), pi(0)) if i '# {"n, . . . , n}. The dynamicsis generated by Ln = An + Sn where An is defined in (2.2.1) and Sn is given by

Sn =n"1!

j="n+1

Y 2j (2.4.7)

Remark that (2.4.5) is also valid for the finite-dimensional dynamics

E'

.!

j

e"a|j|Ej(!(n)(t))

/! C0e

c1t

.!

j

e"a|j|Ej(!)

/(2.4.8)

Choose a initial configuration ! # "a and b > a. Let us define

-n(t) = E'

0$!(n)(t)" !(t)$2

b

1= E'

,!

i#Ze"b|i|

B|r(n)

i (t)" ri(t)|2 + |p(n)i (t)" pi(t)|2

C-

(2.4.9)where the dynamics !(n)(t) and !(t) start from the same initial configuration !.

By Cauchy-Schwarz’s inequality and Ito’s formula, we have

-n(t) ! CT

( t

0

ds-n(s)+CT

!

|i|$n"2

e"b|i|E'

DEi(!) + Ei(!(t)) + Ei(!

(n)(t))E, t # [0, T ]

(2.4.10)where CT is a positive constant depending on T and such that CT goes to zero withT and that can change from line to line. By the a priori bounds (2.4.5, 2.4.8), wehave

E'

DEi(!) + Ei(!(t)) + Ei(!

(n)(t))E! CT ea|i|

,!

j#Ze"a|j|Ej(!)

-(2.4.11)

so that

-n(t) ! CT

( t

0

ds-n(s) + 1n, 1n !CT

b" ae"(b"a)n

,!

j#Ze"a|j|Ej(!)

-(2.4.12)

By Gronwall’s inequality we get

-n(t) ! 1neCT T (2.4.13)

Let Pn the semigroup generated by Ln. Estimate (2.4.13) is enough to provethe following lemma


Lemma 2.4.3 Let a > 0 and ! # "a, then

• P' a.s., (r(n)i (t), p(n)

i (t)) converges to (ri(t), pi(t)) as n goes to infinity.

• For any bounded Lipschitz local function & and t # [0, T ] there exist constantsCn = Cn(a, T, &) ( 0 as n (% such that:

22P tn&(!)" P t&(!)

22 ! Cn

,!

j#Ze"a|j|Ej(!)

-

Let µ and % probability measures on " = (R2)Z, and for any $ , Z let F!

the ensemble of the measurable functions of !! = (rj, pj; j # $) (i.e. functions on" that depends only on the variables in $).

In the following let µ = µgc"0

a grandcanonical Gibbs measure at temperature

#"10 . If % is translation invariant, denoting $n = {"n, . . . , n}, we have that H!n(%|µ)

is a superadditive function of n (see proposition A.1.4), and consequently it existsthe limit

H(%|µ) = limn%&

1

nH!n(%|µ) = sup

n

1

nH!n(%|µ) (2.4.14)

For any local measurable function & define the limit

F (&) = limn%&

1

2nFn(&), Fn(&) = log

(e

Pni=!n !i( dµ (2.4.15)

where $i is the shift operator on functions on (R2)Z.

We recall here proposition A.1.5:

Proposition 2.4.4 Let % a translation invariant probability on ", then

H(%|µ) = sup(

<(& d% " F (&)

=(2.4.16)

where the supremum is taken over all bounded measurable local functions &.

As a consequence we can prove easily the following bound on the averageenergy density: (

E0 d% ! C(1 + H!(%|µ)) (2.4.17)

Here is the ergodic property for the infinite stochastic dynamics:


Theorem 2.4.5 Let % a translation invariant probability measure, stationary forthe dynamics generated by L. Assume that there exist a finite constant C such that

H!(%|µ) ! C|$| (2.4.18)

where µ = µ"0 is a reference grand canonical measure. Then % is a convex combi-nation of grand canonical measures.

We have seen that for a translation invariant %, (2.4.18) is equivalent to thecondition H(%|µ) ! C <%.

The proof is divided in two steps. Let us first consider a generic probabilitymesure %', not necessarily translation invariant, such that H(%'|µ) < %, and let usdenote by g = d)"

dµ . Let us denote, for any n, the Dirichlet forms:

Dn(%') = sup

<"

( Sn(

(d%' : ( # Dom(Sn), ( > 0

=(2.4.19)

A result due to Donsker and Varadhan (proposition A.2.3) says that Dn(%') < +%implies g # Dom(S1/2

n ) and

Dn(%') =n"1!

j="n+1

((Yjg)2

gdµ . (2.4.20)

Proposition 2.4.6 For any probability mesure % and any n, we have

H(%'Ptn|µ) + tDn(%t

',n) ! H(%'|µ) (2.4.21)

where %t',n = t"1

' t

0 %'P snds.

Proof It is a simple consequence of proposition A.2.2 and convexity of theDirichlet form. !

Because of (2.4.20), if H(%'|µ) < +%, then for any j = "n + 1, . . . , n" 1

H(%'Ptn|µ) + t

n"1!

j="n+1

((Yj gt

n)2

gtn

dµ ! H(%'|µ)

where gtn =

d)t",n

dµ . In the second term of the above, because is composed by a sumof positive parts, we can restrict this to any m ! n" 1:

H(%'Ptn|µ) + t

m!

j="m

((Yj gt

n)2

gtn

dµ ! H(%'|µ)


By (A.1.2) and (2.4.19), for any choice of local function & and positive smooth localfunctions (i > 0

(P t

n& d%' " log

(e(dµ" t

m!

j="m

(Y 2

i (i

(id%t',n ! H(%'|µ)

Now we can finally let n (%, and by theorem ?? we obtain

(P t& d%' " log

(e(dµ" t

m!

j="m

(Y 2

i (i

(id%t' ! H(%'|µ) (2.4.22)

where %t' = t"1

' t

0 %'P s ds.

Proposition 2.4.7 Let % be a translation invariant measure stationary for P t suchthat for a finite constant C, H!(%|µgc

"0) ! C|$| for any finite interval $. Then for

any n we have Dn(%) = 0.

Proof We apply now (2.4.22) to %' = %(m)' = %|!m - µ|!c

m. Notice that

H(%(m)' |µ) = H!m(%|µ), and consequently

limm%&

1

2mH(%(m)

' |µ) = H(%|µ)

Choosing & =)m

i="m $i&0, (i = $i(0, where &0 and (0 are local mesurable boundedfunctions, we obtain

m!

i="m

(P t($i&0) d%(m)

' " Fm(&0)" tm!

i="m

($i

Y 20 (0

(0d%(m),t' = H(%(m)

' |µ)

and all we need to prove is that

limn%&

1

2n

n!

i="n

(P t($i&0) d%(n)

' =

(P t&0d% =

(&0d% (2.4.23)

and

limn%&

1

2n

n!

i="n

($i

Y 20 (0

(0d%(n),t' =

(Y 2

0 (0

(0d% (2.4.24)

In fact maximising what we obtained over &0 and ( we get

" inf*0>0

(Y 2

0 (0

(0d% = 0


It is clear that we can repeat the argument substituting Yj to Y0, and we obtain

" inf*0>0

(Y 2

j (0

(0d% = 0

Summing up over j we obtain

0 ! Dn(%) = " inf*>0

n+1!

j="n"1

(Y 2

j (

(d% ! "

n+1!

j="n"1

inf*>0

(Y 2

j (

(d% = 0.

!Proof of (2.4.23) and (2.4.24).

The di!culty comes form the fact that P t& is not local. It is enough to provethat for any i (

P t& d($i%(n)' ) "(

n%&

(P t& d%

This can be done by approximating again P t by our local dynamics P t# . Ob-

serve that by (2.4.17), for any i = 1, . . . , n,(Ej(!)d($i%

(n)' ) ! sup

i#Z

7(Eid% +

(Eidµ

8! K

with K independent of n. Then it follows from Lemma 2.2.1 that

supn

2222(

P t& d($i%(n)' )"

(P t

# & d($i%(n)' )

2222 ! KC#

!

j#Ze"a|j|

with C# ( 0 as * (%.

Now we have that P t# & is a local function, and then

(P t

# &d($i%(n)' ) "(

n%&

(P t

# &d% "(#%&

(P t&d% =

(&d%.

Proof of (2.4.24) is similar.

!

Proposition 2.4.8 If % is a translation invariant probability such that Dn(%) = 0for any n, then %(p|r) is a convex combinations of gaussians probabilities.

Proof The proof basically reduces to the (so called) Poincare Lemma. !We conclude the proof of Theorem 2.4.5, by observing that % is separately

stationary for S and A, and then we conclude by applying Theorem 2.3.3. Observethat the result obtained in proposition 2.4.8 is much more stronger than what weneed to apply theorem 2.3.3 where only exchangeability of %(dp|r) is required.


2.5 Hypoellipticity

In this section, we recall the notion of hypoellipticity and prove that the noiseintroduced in the precedent section is hypoelliptic. Let Zi =

)j Zj

i (x)"xj , i =0, . . . , n be smooth vector fields. We denote by Z'

i its formal adjoint. Let z(x) be asmooth function. We consider operators L of the form

L =n!

i=1

Z'i (x)Zi(x) + Z0(x) + z(x) (2.5.1)

We say that the family of vector fields {Zj} satisfy Hormander condition if theLie algebra generated by the family

{Zi}ni=0, {[Zi, Zj]}n

i,j=0, {[[Zi, Zj], Zk]}ni,j,k=0, . . .

has maximal rank at every point x.

Definition 2.5.1 (Hypoellipticity) A di!erential operator L of the form (2.5.1)is said to be hypoelliptic if it has the regularization property: there exists 1 > 0 suchthat

Lf = g and g # H locs . f # H loc

s++

Theorem 2.5.2 (Hormander) If the family of vector fields {Zj} satisfy Hormandercondition then L defined by (2.5.1) is hypoelliptic.

Corollary 2.5.3 Let L =)n

j=1 Zj(x)'Zj(x) +Z0(x) be the generator of a di!usionand let us assume that (note that Z0 is omitted)

{Zi}ni=1, {[Zi, Zj]}n

i,j=0, {[[Zi, Zj], Zk]}ni,j,k=0, . . .

has maximal rank at every point x. Then L, L', "t"L, "t"L' are hypoelliptic. Thetransition probabilities have smooth densities in space and time and the semigroup isstrong Feller. The invariant measures, if they exist, have a smooth density in space.

We have

Proposition 2.5.4 The di!erential operator)

i#! Y 2i satisfies the conditions of

corollary 2.5.3 on Sn,p,K as soon as |p| </

K.

2.5. HYPOELLIPTICITY 29

Proof We have

[Yj, Yj+2] = (pj"1 " pj)("pj+3 " "pj+2)" (pj+3 " pj+2)("pj!1 " "pj)

[Yj"3, Yj"1] = (pj"4 " pj"3)("pj " "pj!1)" (pj " pj"1)("pj!4 " "pj!3)

and by a simple recursion

[. . . [[Yj, Yj+2], Yj+4], . . . , Yj+2#]

= (pj"1 " pj)("pj+2!+1" "pj+2!

)" (pj+2#+1 " pj+2#)("pj!1 " "pj)

[Yj"2#"1, . . . [Yj"5, [Yj"3, Yj"1], . . .]

= (pj"1 " pj)("pj!2!" "pj!2!!1

) + (pj"2# " pj"2#"1)("pj!1 " "pj)

Let (p1, . . . , pn) # Sn,p,K and assume that |p| < K (otherwise the manifold isreduced to a single point). Observe that the vector fields "pj satisfy

!

j#TN

"pj = 0,!

j#TN

"pj = 0

Then there exists an index j # 1, . . . , n" 2, say j = 1, such that pj+1" pj '= 0and ("p3 , . . . , "pn) is a basis of the tangent space to Sn,p,K at point (p1, . . . , pn).

Observe now that

Rank({(p1 " p2)("p!+1" "p!

)" (p#+1 " p#)("p1 " "p2), * = 1, . . . , n" 2})= Rank({"p3 , . . . , "pn}) = n" 2

because the condition (p2 " p1) '= 0. It follows that the Lie algebra generated by{Y2, . . . , Yn"1} is the full tangent space to the (n" 2)-dimensional sphere

Sn,p,K =

%(p1, . . . , pn) # Rn :

n!

j=1

pj = np,n!

j=1

p2j = nK

&.

at every point. !

Proposition 2.5.5 The infinitesimal generator Ln = An + 0Sn on the surface ofconstant energy

Hn,p,K =

%(ri, pi)i#Tn ;

!

i#TN

Ei = nK,!

i#TN

pi = p

&, |p| <

/2K

satisfies conditions of corollary 2.5.3.

Proof No it is false. If &j, pj = p with |p|2 < 2K then &j, Yj = 0 so the Liealgbra is generated by

)j(V

!(rj+1)" V !(rj))"pj which is not of maximal rank ! !


2.6 Other stochastic dynamics

2.6.1 Energy conserving noise

We can also define a noise S =)

j X2j acting on the momenta conserving only

kinetic energy. The construction is similar : for nearest neighbors atoms j, j + 1 wedefine the vector field Xj by

Xj = pj+1"pj " pj"pj+1 (2.6.1)

It is tangent to the circle {(pj, pj+1) # R2; p2j + p2

j+1 = 1} so that

S =!

j

X2j (2.6.2)

conserves the kinetic energy. Momentum is not conserved and is in fact eigenvectorof S since S(pj) = "pj.

2.6.2 Momentum exchange and momentum flip

One can also consider noise of Poissonian type conserving energy and eventuallyalso momentum. Poissonian energy conserving noise is defined by the following flipoperator

(Sf)(p) =!

j

Df(pj)" f(p)

E(2.6.3)

where pj is the configuration obtained from p by changing the coordinate pj in "pj.

Poissonian momentum-energy conserving noise is realized by exchange of mo-menta of nearest neighbor atoms. The generator of this noise is given by

(Sf)(p) =!

j

Df(pj,j+1)" f(p)

E(2.6.4)

where pj,j+1 is the configuration obtained from p by exchanging the coordinates pj

and pj+1.

Remark that these noise have very poor ergodic properties (which is not thecase of the Brownian noises defined before). Nevertheless results of section 2.3 canbe applied for them and it implies that the dynamics obtained by adding these noisesto the Hamiltonian dynamics is ergodic.

2.7. BIBLIOGRAPHICAL NOTES 31

2.7 Bibliographical Notes

The argument for the proof of the ergodicity of the stochastic model is adaptedfrom [11], [10], [5] and [6].


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Chapter 2 Dynamics