A CONSERVATIVE-DISSIPATIVE DYNAMICS FOR A SPIN SYSTEM
THIERRY GOBRON AND LIVIO TRIOLO
Abstract. We introduce a model of stochastic dynamics in which the nonlinear operator
for the evolution of states consists of two parts: a Markov term describing a stochastic
conservative (i.e. iso-energetic) evolution and a nonlinear term which allows for energy
fluctuations by increasing the entropy of the distribution We study this dynamics in the
case of a simple one-dimensional Ising spin system. Interest in such a model lies in the
fact that the dynamics is not isothermal at variance with Glauber-type dynamics. Here a
notion of inhomogeneous temperature field naturally arise in the system through energy
fluctuations. When time goes to infinity, the distribution is shown to converge to a
canonical distribution with a well defined temperature related to the initial mean energy.
In the limit of strong dissipation, a reduced dynamics can be defined on a manifold of
generalized Gibbs states.
1. Introduction
Stochastic Ising models[G] have been introduced already quite a while ago in order to
describe qualitatively the relaxation process of a statistical system coupled to a thermal
bath. Such models have contributed in a large part to the construction of statistical
mechanics out of equilibrium and have given (and still give) an enormous variety of results,
such as equilibrium fluctuations, metastability, interface dynamics, and so on. Maybe the
reasons for their success come from a twofold interest: from a physical point of view,
and for a rather wide variety of circumstances, the basic assumptions on the coupling
with a thermal bath may be considered as reasonable; this makes possible the reduction
from (quantum) Hamiltonian to stochastic dynamics [M]; from a mathematical point of
view, such models have the structure of a Markov processes, and this very important
feature allows the interplay with probability theory to a very large extent without further
structural reduction.
The situation is rather different when one wants to consider models in which thermal
equilibrium is not assumed from the outset but is a part of the dynamical process to
be described. We are still lacking a simple model which could close the gap between
microscopic models (with some “true” dynamics) and more phenomenological ones, such
as phase field models, which are mostly based on thermodynamics assumptions. In this
paper we introduce a model of dynamics which may be seen as a first step in that direction.
1991 Mathematics Subject Classification. Primary; Secondary .
Key words and phrases. Nonequilibrium Statistical Dynamics, Dissipative Dynamics, Relative Entropy
Methods, Generalized Gibbs States.
Partially Supported by CNR and MURST.
1
2
THIERRY GOBRON AND LIVIO TRIOLO
The starting point for the present construction consists in noting that the widely used
reversibility with respect to a Gibbsian state for a Markov process cannot account for non
isothermal dynamics because the parameters of the equilibrium distribution are already
encoded in the rates. One should expect different canonical equilibria starting from different initial conditions for the same (isolated) system. Only conservative or micro-canonical
processes can be considered as an exception to this, and a notion of temperature arises
from the fluctuations of large subsystems. However, these fluctuations are those of the
system itself, arising through rescaling, and not due to other degrees of freedom as in the
case of the existence of a heat bath. As a matter of fact, we are lacking some insights on
what should be a reasonable reduced dynamics in the case when the thermal bath is not
strong enough, and we propose here the simplest non-Markovian modification which could
account for such effects: we first consider a Markovian generator for a micro-canonical
dynamics which describes the evolution of the energy distribution and add to it a term
which should account for the fluctuations due to irrelevant degrees of freedom. Mostly for
simplicity, we introduce a BGK-like term [BGK] which is essentially a drift term toward
the distribution of maximal entropy with the same mean profile for the order parameter
and the energy.
For small values of this drift term, it allows only for some fluctuations around the microcanonical distribution, but has strong effects for instance on the equilibrium distribution
which we prove to be a canonical distribution even in finite volume, with a temperature
related to the initial mean energy.
For large values of the drift term, the physical relevance of such a model is certainly more
questionable, since fluctuations get much larger than the supposed main process. However,
it allows us to make a clear connection with other approaches [S], where a constrained
entropy is maximized at each time. We thus define generalized Gibbs states as the points
of maximal entropy with constraints and show that a reduced dynamics can be defined on
the related manifold and correctly describes the behavior of the system already for finite
but large value of the drift term.
The paper is organized as follows. We first give some definitions and recall some relevant
preliminary notions on the entropy maximization, introducing the manifold of “generalized
Gibbs states”. In the next section, we introduce the “conservative-dissipative” dynamics in
the state space and we study its asymptotic behavior for large time, proving the convergence
to a canonical Gibbs state with a well defined temperature related to the initial mean total
energy. In the last section we study the case of large dissipation and show by relative
entropy methods the stability of the reduced (asymptotic) dynamics.
2. Preliminaries and Definitions
In this section we introduce various definitions and notations which will be useful in the
sequel. We consider a finite system of N spins whose 2N configurations are
(2.1)
s = {s1 , s2 , ...sN } , sx ∈ {−1, 1} , x = 1, 2, ...N
CONSERVATIVE-DISSIPATIVE DYNAMICS
3
and we denote the whole space of configurations as
Ω = {−1, 1}N
(2.2)
Let M1 + denote the set of states ρ on Ω (positive normalized measures): i.e.
X
(2.3)
ρ : Ω → [0, 1],
ρ(s) = 1
s∈Ω
In a more geometrical picture, M1 + identifies with the unit simplex in R2 , which vertices
are the 2N points at distance one from the origin on the 2N positive semiaxes, and can be
made in one to one correspondance with the 2N atomic measures on the configurations,
X
X
αc = 1
(2.4)
ρ(s) =
αc 1{c} (s), αc ≥ 0,
N
c∈Ω
c∈Ω
where 1A (·) denotes the indicator function of the set A. We also recall the usual definition
of mean value of an observable f : Ω → R:
X
f (s)ρ(s)
ρ(f ) :=
s∈Ω
In the next sections, we will consider the relative entropy between two states ρ and ν in
M1 + , which is defined as
X
ρ(s)
ρ(s) log[
(2.5)
η(ρ|ν) =
]
ν(s)
s∈Ω
◦
which is strictly positive whenever ρ 6= ν and finite for ρ and ν in the interior M1 + of the
simplex M1 + . Furthermore, it gives a convenient upper-bound for the L1 distance between
states, by the Kullback-Leibler inequality [LR],
kρ − νk21
2
where the L1 distance is defined as,
(2.6)
(2.7)
for all ρ and ν in M1 +
η(ρ|ν) ≥
kρ − νk1 =
X
|ρ(s) − ν(s)|
s∈Ω
We also recall the notion of Boltzmann-Gibbs entropy, as a function on M1 + : it is given,
up to a multiplicative positive constant, by
X
(2.8)
η(ρ) = −
ρ(s) log ρ(s)
s∈Ω
In the following we will be interested in various states which are constrained maxima of the
entropy η. We consider the (ferromagnetic) Ising Hamiltonian on this system with nearest
neighbors interactions and unit coupling constant,
(2.9)
H(s) = −
N
X
x=1
sx sx+1
4
THIERRY GOBRON AND LIVIO TRIOLO
In this equation and wherever needed for notational convenience, we identify sN +1 and s1 .
We will denote by µ̄β the related canonical state at inverse temperature β
N
(2.10)
X
1
1
exp(β
sx sx+1 )
exp(−βH(s)) =
µ̄β (s) =
Zβ
Zβ
x=1
(2.11)
Zβ =
X
s∈Ω
N
X
exp( (βsx sx+1 ))
x=1
µ̄β is the unique maximum for the entropy functional on M1 + under the constraint of a
fixed mean value for H,
(2.12)
η(µ̄β ) = max{η(ρ); ρ(H) = E}
It should be noted that, with the Hamiltonian given in (2.9), the parameter β takes positive
values (and hence can be directly interpreted as an inverse temperature) only if the mean
energy E is non-positive, which is consistent with the situation of ferromagnetic interactions
and justify the sign in the right hand side of (2.9). The positive values of E have to be
discarded here, since obviously they do not correspond to any reasonable equilibrium state
for ferromagnetic interactions. From a mathematical point of view, the maximization
procedure is still possible, leading to negative values for β, in accordance with the weird
feature that the number of accessible states would decrease with energy. A more standard
thermodynamics would be recovered by considering anti-ferromagnetic interactions, for
instance reversing the global sign of H in (2.9). For our present purpose, we need to
generalize this maximization procedure and introduce the generalized Gibbs states in the
following sense.
For any state ρ in M1 + , we denote its one-point and two-(nearest)-point correlation
functions in x as the mean values of the particular observables sx and sx sx+1 , i.e.
X
X
sx ρ(s) , ρ(sx sx+1 ) =
sx sx+1 ρ(s)
(2.13)
ρ(sx ) =
s∈Ω
s∈Ω
The unique state which maximizes the entropy functional, under the constraints of fixed
values for all the one-point and two-(nearest)-point correlation functions will be called a
generalized Gibbs state. We first need the following
Definition 1. (m, e) in RN × RN is an admissible magnetization-energy profile for the system, if there exists a positive constant δ such that m = (mx )x∈{1,··· ,N } and e = (ex )x∈{1,··· ,N }
verify the following inequalities for all x in {1, · · · , N },
(2.14)
1 − |mx | > δ
1 + ex − |mx + mx+1 | > δ
1 − ex − |mx − mx+1 | > δ
These conditions allow us to interpret m and e as sets of one-point and two-point correlation functions.
From convexity of η(·) and linearity of the constraints, the following result can be derived:
CONSERVATIVE-DISSIPATIVE DYNAMICS
5
Proposition 1. Let (m, e) be an admissible magnetization-energy profile. Then there exists
◦
a unique state µ ∈ M1 + such that
(2.15)
η(µ) = max η(ρ); ρ(sx ) = mx , ρ(sx sx+1 ) = ex , x ∈ {1, · · · , N }
This state can be parametrized by 2N real constants, h = (hx )x∈{1,··· ,N } and β = (βx )x∈{1,··· ,N } ,
and written as
N
X
1
(2.16)
µh,β (s) =
exp( (hx sx + βx sx sx+1 ))
Zh,β
x=1
with the normalization
(2.17)
Zh,β =
X
N
X
exp( (hx sx + βx sx sx+1 ))
x=1
s∈Ω
and there is a one-to-one correspondence between the values of the constraints m, e and
the parameters β, h.
We call such states “generalized Gibbs states”. The set ΓG of generalized Gibbs states,
through the (h, β)- parametrization gets the structure of a regular 2N -dimensional manifold in M1 + , and its closure Γ̄M contains the atomic measures supported by a single
configuration. Proposition 1 can be extended to infinite systems under the condition that
the constant δ which appears in (2.14) remains bounded away from 1 in the limit N → ∞,
or equivalently that all |hx | and |βx | are uniformly bounded in x [COE]. The canonical
states associated to the Hamiltonian (2.9) and inverse temperature β are in ΓG and correspond to the case hx = 0, βx = β > 0, x ∈ {1, · · · , N }. one-point and two-point correlation
functions.
The correlations can be expressed for a generalized Gibbs state µh,β , as
(2.18)
mx = µh,β (sx ) =
sx µh,β (s) =
∂ log Z(h, β)
∂hx
sx sx+1 µh,β (s) =
∂ log Z(h, β)
∂βx
X
s∈Ω
(2.19)
ex = µh,β (sx sx+1 ) =
X
s∈Ω
In the case studied here, the inverse formula is explicit in the limit N → ∞, as
(1 − mx )2 (1 + mx )2 − (mx−1 + ex−1 )2
(2.20)
exp (4hx ) =
(1 + mx )2 (1 − mx )2 − (mx−1 − ex−1 )2
(1 + mx )2 − (mx+1 + ex )2
×
(1 − mx )2 − (mx+1 − ex )2
(1 + ex )2 − (mx+1 + mx )2
(1 − ex )2 − (mx+1 − mx )2
One may note that the right hand side of the equations (2.20) factorizes in fractions which
are bounded away from 0 and +∞ as a consequence of the inequalities (2.14). This
(2.21)
exp (4βx ) =
6
THIERRY GOBRON AND LIVIO TRIOLO
guarantees the uniform boundedness of |hx | and |βx |. These results are still valid for finite
N up to an exponentially small error term. We also need to extend P
the notion of microcanonical states. The admissible values of the total energy H(s) = − x sx sx+1 are easily
seen to be
N
Ej = −N + 4j, j = 0, · · · , [ ]
2
where [·] denotes the integer part. All these energy levels are degenerate; for j = 0, · · · , [ N2 ]
let
Σj := H −1 (Ej ) = {s : H(s) = Ej }
(2.22)
Let νj be the micro-canonical state corresponding to the energy Ej ,
1 X
1{c} (s)
(2.23)
νj (s) = N 2 2j c∈Σj
Equivalently νj is the uniform measure on the set Σj .
Let us finally denote by ΓM the set of generalized micro-canonical states which we define
as the convex combinations of the νj ,
X
X
(2.24)
ΓM := {ν : ν =
αj νj , αj ≥ 0,
αj = 1}
j
j
3. The Dynamics
In this section, we construct a dynamics on M1 + . It converges to the canonical state µ̄β ,
where the inverse temperature β is determined by the initial value of the total energy ρ0 (H)
◦
for all initial states ρ0 in M1 + . This dynamics is constructed as a BGK-type modification
[BGK] of a conservative linear Markovian dynamics, as
∂ρ
= Lρ + κ(µ[ρ] − ρ)
(3.1)
∂t
◦
for any initial state in M1 + . The two terms in the right hand side of the above equation
are defined as follows.
In the first term of (3.1), L is a linear Markov generator which preserves the energy H,
and acts on ρ as,
X
(3.2)
Lρ(s) = λ
(1 − sx−1 sx+1 )[ρ(sx ) − ρ(s)]
x
+
for all ρ in M1 and where, as usual
(3.3)
(sx )y = sy if x 6= y ,
(sx )y = −sy if x = y
λ is a positive constant which fixes the rates of spin flips.
In the second term of (3.1), we define the state µ[ρ] as the state of maximum entropy with
the same one-point and two-nearest-point correlations as ρ. It follows from Proposition 1
◦
that µ[ρ] is uniquely defined provided ρ is in M1 + . κ is a real parameter which fixes the
intensity of the dissipative term.
CONSERVATIVE-DISSIPATIVE DYNAMICS
7
By convexity properties of M1 + and Proposition 1, the dynamics is well defined for all
finite time and all initial state in the interior of M1 + . A first property of this dynamics is
that the mean value of the energy ρ(H) is constant in time since H is preserved by L and
has the same mean value for ρ and µ[ρ],
X
X
(3.4)
µ[ρ](H) = −
µ[ρ](sx sx+1 ) = −
ρ(sx sx+1 ) = ρ(H)
x
x
This leads directly to
∂ρ(H)
=0
∂t
In this model, the two point correlation functions ρ(sx sx+1 ) evolves in a diffusive way at a
rate fixed by λ,
∂ρt (sx sx+1 )
(3.6)
= λ ρt (sx−1 sx ) + ρt (sx+1 sx+2 ) − 2ρt (sx sx+1 )
∂t
The one-point correlation functions ρ(sx ) have however a different behavior, since magnetization is not preserved by the linear Markov term,
∂ρt (sx )
(3.7)
= −λ ρt (sx ) − ρt (sx−1 sx sx+1 ) )
∂t
This second set of equations is not closed but depend on higher order terms. Another
important property is that under this dynamics, any state converges to a canonical Gibbs
distribution. The associated inverse temperature is fixed by the conservation of the mean
value of the energy. Let ρt be the evolved state at time t starting from ρ0 , and E the
initial value for the mean energy, E = ρ0 (H). We denote β(E) the value for the inverse
temperature such that the canonical state has mean energy E, µ̄β(E) (H) = E.
(3.5)
Proposition 2. There exists a constant cN > 0 such that for any initial state ρ0 in the
interior of M1 + , there exists a constant K such that,
kρt − µ̄β(E) k1 ≤ K exp {−cN t}
(3.8)
for all time t > 0
Proof: We consider the time derivative of the relative entropy of ρt with respect to
µ̄β(E) . We have,
∂η(ρt |µ̄β(E) )
λ X X 1 − sx−1 sx+1
ρt (sx ) − ρt (s) ln(ρt (sx )) − ln(ρt (s))
= −
2
∂t
2 s x
X
(3.9)
−κ
µ[ρt ](s) − ρt (s) ln(µ[ρt ](s)) − ln(ρt (s))
s
For every state ρ we denote by ν[ρ] the state in ΓM which is a superposition of microcanonical states with the same weights as ρ on each energy level. We thus have,
∂η(ρt |µ̄β(E) )
λ
(3.10)
≤ − η(ρt |ν[ρt ]) − κη(ρt |µ[ρt ])
∂t
2
8
THIERRY GOBRON AND LIVIO TRIOLO
Convergence to µ̄β(E) follows from the fact that it is the intersection point between the
set of generalized Gibbs states with mean energy E, and the set of of states which are
superpositions of micro-canonical states, and have the same mean energy E. Exponential
convergence to zero of the relative entropy follows from transversality of the two sets, and
L1 convergence from Kullback-Leibler inequality.
4. Large Dissipation and Reduced dynamics
In the case of a large value of the dissipation constant κ, one expects that the evolution
fastly drives the state to a close neighborhood of the manifold ΓG . The later convergence
to the equilibrium state could be traced through a simpler (reduced) dynamics defined on
ΓG . In the preceeding section, we stressed the fact that the evolution equations for the
correlation functions are not closed. However, in the case of large dissipation (i.e. for
large values of κ), The state of the system ρt evolves toward a small neighborhood of the
manifold ΓG in a time of order κ1 and it is expected that its subsequent evolution can be
well approximated by a reduced dynamics defined on ΓG .
◦
Proposition 3. For each initial state ρ0 in M1 + , there exists a constant C(ρ0 ) ≥ 0 such
that for all t,
C(ρ0 )
(4.1)
η(ρt |µρt ) ≤ η(ρ0 |µρ0 ) exp −κt +
κ
Proof: The main step consists in establishing the following differential inequality, which
is straightforward,
X
∂dη(ρt |µρt )
(4.2)
≤λ
(βx (ex+1 − 2ex + ex−1 ) + hx (mx − tx )) − κη(ρt |µρt )
∂t
x
◦
Since the values βx and hx are bounded uniformly in time for each initial state ρ0 in M1 + ,
the first term in the right hand side of the above equation is bounded by a constant C(ρ0 ).
Integration of the resulting inequality gives then the result.
Let h = h(m, e) and β = β(m, e) be the parameters of the state in ΓG associated to
the admissible profile (m, e). We denote by τx (m, e) its three-(adjacent) point correlation
function at site x,
(4.3)
τx (m, e) = µh,β (sx−1 sx sx+1 )
In the limit of large N , τx (m, e) has the following explicit expression,
(4.4)
τx (m, e) =
mx−1 ex + mx+1 ex−1 − mx (mx−1 mx+1 + ex−1 ex )
1 − m2x
The following equations,
(4.5)
∂mx
= −λ(mx − τx )
∂t
(4.6)
∂ex
= λ(ex+1 − 2ex + ex−1 )
∂t
CONSERVATIVE-DISSIPATIVE DYNAMICS
9
define an evolution for the profile (m, e). This provides the reduced dynamics on the
manifold ΓG .
5. Acknowledgements
L.T. and T.G. thank respectively the Université de Cergy-Pontoise and the Università
di Roma Tor Vergata for their kind hospitality during various stages of this collaboration.
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THIERRY GOBRON AND LIVIO TRIOLO
LPTM, Université de Cergy-Pontoise, Cergy-Pontoise, France
Dipartimento di Matematica, Università di Roma Tor Vergata, Roma Italia
E-mail address: [email protected]
E-mail address: [email protected]