Thermodynamics in Terms of a Sequence of $n-$chains Derived from a Martingale Decomposition of the Energy Process

0.26 MB·English

by  David ford

Thermodynamics in Terms of a Sequence of n−chains Derived from a
Martingale Decomposition of the Energy Process
6
0
0 David Ford∗
2 Department of Physics, Naval Post Graduate School, Monterey, California
(Dated: February 6, 2008)
n
a
The role of the algebraic method has long been understood in shedding light on the topological
J
structure of sets. However, when the set is a simplicial complex and host to a dynamical process,
1
in particular the trajectory of a canonically distributed system in thermal equilibrium with a heat
3
bath,the algebra re-enters. Via a theorem of L´evy and Dynkin,there is a correspondence between
a system’s energy process at equilibrium and a sequenceof n−chains on the state space.
]
h
c
e FUNDAMENTALS AND REVIEW where,
m
- Thermostatics
t
a N1
t Π= p1p2···pN (2)
s Lettheset{ω1,ω2,...,ωN}bethestatespace (cid:16) (cid:17)
.
t for a random process {xt : t ∈ [0,∞)}. Accord-
a and for k =1,2,...,N
ing to the results of [1], under the assumption
m
that the “position” process x is a trajectory of
t
d- a canonically distributed subsystem in equilib- pk = p(xt ∈ωk) .
n rium with a heat bath at temperature θ, there
o is a well defined energy process Further [2], if the N−state process x has a
t
c
characteristic Carlson depth (in the finite state
[
Markoff case this is simply the L1 norm of the
Π
1 h(x )=θ log (1) cycle between visits to a rare state) we have
t
v (cid:16)p(xt)(cid:17)
6
1
7
1
const.
0 θ(∆t)= (3)
6 ktk (log[ Π˘ ])2+(log[ Π˘ ])2+···+(log[ Π˘ ])2+1
0 2q ∆t1 ∆t2 ∆tN−1
/
t
a
m
where, ∆t is the characteristic time associated ternal energy
- k
d with the kth state during a cycle and
n U =p·h
o
1
c Π˘ = ∆t ∆t ···∆t N . willbeofparticularimportanceinthenextsub-
: 1 2 N
v (cid:16) (cid:17) section.
i
X
For a simple example see [3]. Sample Isotherms
r asseenfromthetimedomainareshowninfigure Thermodynamics
a
1.
With these microscopic quantities in hand, It may happen that there are changes to the
one is in a position to calculate the free energy observed probabilities, changes in the length of
and many other important quantities. The in- the characteristic time scale or both. if it is as-
2
Frequently,asthemicroscopicdynamicsoften
occuronstatespaceswithatopologythesesame
macroscopic phenomena may also admit a sec-
ond, low-dimensional interpretation in terms of
their interplay with features of the state space.
The purpose of this paper is to illustrate the
twofold nature of the thermodynamics. The
connection is accomplished via the subsystem’s
trajectory through its path space and the path
decomposition of L´evy and Dynkin.
Dynkin-Le´vy Decomposition of the Path
FIG.1: SurfacesofConstantTemperatureforaTwo This subsection focuses some elementary as-
State System pects of the martingale theory on applications
totheenergyprocess. Inparticular,thecontext
is that of a canonically distributed subsystem’s
sumedthatthesechangesareoccurringinasub- trajectory through its state space. For readers
systemthatisinastateofcontinualequilibrium unfamiliar with the martingale point of view an
witha heatbath(orperhaps the observershifts excellent reference for this section is [4].
her attentionfromone subsystemto anotherall Recall that according to the L´evy Formula,
in established equilibrium with the same bath), see for example [5], the martingale decomposi-
the notions ofworkdone,heatgainedorlostby ton [4] of a Markoff process is given by:
the system become applicable.
In equilibrium these changes are given by the t
productrulefordifferentiation(theFirst Law) g(xt)−g(x0)−Z qxsk(g(k)−g(xs))ds
as 0 kX6=xs
here
U˙ = p·h˙ + p˙ ·h (4) q = lim P(xs+ =k |xs)
xsk s+→s s+−s
workrate heatrate
|{z} |{z} is the generator of the process.
In the nonstationary case equation (2) trans-
Ofparticularinterestinthe sequelis the case
formsintheobviouswayfromΠintoΠ(t). Sim-
wheng(·)istheenergyfunctionh(x ,t)andthe
ilarly h(x ) becomes h(x ,t), etc. The calcula- t
t t
temperature is held constant. Under these con-
tionoftheratesinvolvetimederivativesofthese
ditions the Dynkin-L´evy kernel becomes
quantitiesandareeasilycomputed. Thevectors
appearing in the dot products on the RHS of Π(s+) Π(s)
equation (4) are typically high dimensional. θ qxsk(s) (cid:18)logp(k,s+) −logp(x ,s)(cid:19).
Fortunately,inthestatisticalmechanicalcon- kX6=xs s
text, the work rate admits a low dimensional Note: it may also be that the generator is itself
description as well. Typically this is system de- a function of time.
pendent. Famousexamplesincludepressureand
volumechangesormagnetizationandchangesin
an applied field. The low-dimensional descrip- On the Choice of State Space
tion of the heat rate is often system indepen-
dent and is given in terms of temperature and As the main purpose of this paper is to bring
entropy. outtheconnectionbetweenthephysicalandand
3
algebraicdescriptionsofthe subsystem’s trajec- The rows represent the states accessible to the
toryitisworthwhiletochooseastatespacethat system and may be used as state labels.
is meaningful in either context. For an instantiation based on M balls and m
Exchange rules in the context of urn models, urns,the cardinalityofthe statespaceissimply
[6], [7], [8], are frequently used as state spaces the number of ways to put M balls (indistin-
to study diffusion, spin flip dynamics, granular guishable) in m urns. In equation (5) an ex-
materials,phasetransitions,etc. Furtherthere- ample of the state space matrix and row label-
sulting state spaces are often simplexes or sim- ing scheme are presented. The left hand side of
plicial complexes which have long been used as equation (5) defines the list of state labels as-
building blocks in the study of algebraic geom- sociated with a space based on three urns and
etry and topology. threeballs(atotaloftenstates). Therighthand
Aparticularlysimplememberofthisfamilyof side of the equation lists the states themselves.
exchangemodelsistheGordon-Newellqueue[9]. The dimension of the column space is equal to
Foraninterpretationofthatprocessintermsof the number of urns.
its thermodynamics see [1]. Figure 2 illustrates In the sequel there will be essentially two
the algebraic topology associated with its dy- types of vector dot products, those describing
namics. Despite the potential for an arbitrarily macrostructureandthosedescribingmicrostruc-
large number of non-degenerate (energy levels ture. For clarity, dot products with respect to
are distinct) microstates, both the thermody- the low dimensional row space will be denoted
namic and algebraic descriptions of the energy with a circled circle notation, for example (see
process are three dimensional. The connection equation (5))
will be more fully developed in the sequel.
ααα(7)⊚ηηη =1η +2η +0η .
1 2 3
Dotproductsinthecolumnspaceandotherhigh
dimensional vectors will use the standard “ ··· ”
notation.
The labels ααα(1),ααα(4) and ααα(10) are the
0−simplexes. These are the states with all the
ballsinoneurnor,alternatively,thestateswith
two urns empty. The states ααα(2) and ααα(3) be-
long to the same 1−simplex, namely the set of
states with the first urn empty and the others
non-empty. The labelααα(6) state belongs to the
interior of the region. That is, the set of states
FIG. 2: A basic example of a 2−simplex derived with no empty urns.
from the state space for Gordon-Newell exchange.
The 0−chain, 1−chain and 2−chain shown (color
coded) are derived from the martingale decomposi- ααα(1) 0 0 3
tion of the local energy process. ααα(2) 0 1 2
ααα(3) 0 2 1
ααα(4) 0 3 0
ααα(5) 1 0 2
= (5)
ααα(6) 1 1 1
Elements of the Gordon-Newell Process ααα(7) 1 2 0
ααα(8) 2 0 1
ααα(9) 2 1 0
The state space of a Gordon-Newell process
ααα(10) 3 0 0
may be represented as a matrix. The dimen-
sion of the column space is the number of urns.
3 urns
| {z }
4
Abasicversionoftheexchangeruleconsistsof of the exchange rule parameters and was intro-
anm×mglobalroutingtable(atransitionprob- ducedinthelastsection. Themiddleexpression
abilitymatrixforanembeddeddiscreteprocess) follows using the substitution
and m−timescales that characterize (globally)
the rate of activity of the individual urns. π
η =−log( .) .
Let Pij denote the probability that the dis- . q.
crete process transitions a single ball from urn
“i” to urn “j”. The stationary vector (eigen-
The bottom expression gives a stationary mea-
value1)ofthetransitionmatrixwillbedenoted
sure in terms of energy and temperature.
{π ,π ,...,π }. The urn rates will be denoted
1 2 m
{q ,q ,...,q }. Asisapparentfromequation(3),thetemper-
1 2 m
Anycomponentofastationarymeasureofthe aturemayvary(viadilatationofthetimescales)
process may be given in terms of these parame- even though the state probabilities themselves
ters: are constant in time. When both the tempera-
ture and probabilities are constant in time
π π π
( 1)α1( 2)α2···( m)αm
q q q
1 2 m
h(x )
t
where, {α1,α2,...,αm} are entries in that row h(xt,t)= (7)
of the state space matrix associated with the θηηη⊚(ααα(xt)−ααα¯)
state labelααα(·). 
A slightly more flexible version of the ex-
where ααα¯ is the arithmetic mean of the rows of
change rule allows the routing table and rates
the state space matrix. For the example given
to vary locally from state to state. In addition,
in equation (5) this vector is {1,1,1}.
the parameters may be functions of time.
Dynkin-L´evy kernel for the energy process is
given by
THE CONNECTION WITH THE HEAT
RATE PROCESS
ηηη⊚(ααα(x )−ααα(k))
The Constant Temperature, Time s
Stationary Equilibrium qxsk θ (8)
kX6=xs  h(ααα(xs))θ−h(ααα(k))
With the state energies and temperature 
given in terms of the canonically distributed InwhatfollowstheDynkin-L´evycoefficientata
subsystem’s statistics and timescales by equa- site “∗” will be denoted dl(∗).
tions (1) and (3), a stationary measure µµµ(·) of
theexchangeprocessmaybeexpressed(compo-
nentwise) in a few ways
Physical Significance of the Dynkin-Le´vy
(π1)α1(π2)α2···(πm)αm Kernel for the Energy Process
q1 q2 qm

µ(ααα)= e−(η1α1+η2α2+...+ηmαm) (6) staUnntdteermtpheerahtyuproe,thteimsiseostfatthioisnasurybseeqcutiiolinb,ricuomn-,
 e−h(θααα). tehqeuaDtiyonnki(n8-)L.´eIvtymkaerynbeletsahkoewsnthtehfaotrmthigsivceonrrien-
Recall that the top expression on the RHS of sponds to the heat process associated with the
equation (6) is the stationary measure in terms energy process h(x ). For
t
5
TABLEI:Abreakdownofthesummandsthatcom-
hdl(x )ids = p(x ,s)P (s,s+) prise a Dynkin-L´evy coefficient: possible transition
s s xsk states,α−indexdifferencesandcorrespondingsmall
Xxs Xk time probabilities of transition. Seefigure 3.
×[ααα(k)−ααα(x )]⊚ηηη
s
= p(k,s+)[ααα(k)−ααα¯]⊚ηηη k α(k)−α(A) pAk(s,s+)
A 0,0,0 1 - ( )
Xk B 1,0,−1 q3P31 ds
− p(xs,s)[ααα(xs)−ααα¯]⊚ηηη C 0,1,−1 q3P32 ds
Xxs D −1,1,0 q1P12 ds
E −1,0,1 q1P13 ds
= (p(m,s+)−p(m,s)) F 0,−1,1 q2P23 ds
Xm G 1,−1,0 q2P21 ds
×[ααα(m)−ααα¯]⊚ηηη .
So that,
hdl(x )i = p˙(m,s)h(m,s) . (9)
s
Xm
This is of course the heat rate’s contribution to
the “First Law”, equation (4).
Algebraic Significance of the Dynkin-Le´vy
Kernel for the Energy Process
Inspection of equation (8) shows that the
FIG. 3: Typical Neighborhood of an Interior Point
Dynkin-L´evycoefficientatasitedependsonthe in the StateSpace
ααα differences with the states to which the pro-
cess may transition, weighted by the probabil-
ity rate that the transition takes place during a
small time s→s+. Consider the 1−simplex of states that have
Recall from the previous section that in the the first urnempty andthe other two urns non-
most basic version of the exchange rule there is empty (for a picture see the orange 1-boundary
one global matrix (site independent) that gov- in figure 2). The typical neighborhood and ta-
erns the transitions of the embedded discrete ble for this boundary are obtained from those
process and one set of inverse characteristic shown above for the interior points by simply
times that governs the rates of the urns. deleting points “D” and “E”. The same table
A typical local neighborhood of an interior and figure result for all points on this face. In
point “A” with state label ααα(A) = {a,b,c} is this way a real valued coefficient is associated
shown in figure 3. The quantities relevant to witheach1−simplexinthe complex. The setof
the calculation of the Dynkin-L´evy coefficient these coefficients specifies a particular 1−chain.
at site “A” are listed in the table. The reader should be able to convince her-
Noticethatthetableentriesareinvariantwith self that under the conditions of the basic ex-
respect to the actual location of site “A”. That changerule: one global(site independent) tran-
is, the Dynkin-L´evy coeficients are constant on sitionmatrixandoneglobalsetofcharacteristic
the interior of the simplex. A similar situa- times for the urns, the Dynkin-L´evy coefficients
tion holds in the case “A” belongs to one of the fortheenergyprocessspecifyn−chainsinhigher
boundaries. dimensions as well.
6
THE CONNECTION WITH THE WORK Underthesameconditions,inthecaseofcon-
RATE PROCESS stanttemperaturebuttimevaryingdynamics,it
is shown that, due to the presence of a site de-
It may happen, see equation(3) or figure 1, pendent (local) work rate term, the correspon-
that the probabilities vary in time but the sys- dence with the chain groups is lost. Thus, the
temremainsonaconstanttemperaturesurface. connection has been made between transitions
Inthissituationthequantitiesπ,q andη from from one thermal equilibrium to another and
... ... ...
equation (6) become functions of time. The fromonesequenceofp-chainsonthestatespace
Dynkin-L´evy kernel of the energy process takes to another.
the form (see the appendix, equation (18) )
APPENDIX A: DERIVATION OF THE
dl(xs) = − qxsk(s) DYNKIN-LE´VY KERNEL FOR THE
kX6=xs ENERGY PROCESS
× h(ααα(x ),s)−h(ααα(k),s+)
s
(cid:0) (cid:1) Let h(x ,t) denote the energy of the state x
t t
= θηηη˙(s)⊚(ααα(x )−ααα¯¯¯) at the time t. That is, the Hamiltonian for the
s
system may have an explicit time dependence.
+θ q (s)
xsk Thecollectionofallsystemtrajectories(thepo-
kX6=xs sitionprocesses{x ,t∈[0,∞)})willbedenoted
t
×ηηη(s)⊚(ααα(k)−ααα(xs)) Ω. Let P denote the probability on path space
(10) thedescribesthetrajectoriesofthepositionpro-
cess.
Comparison of (10) with equation (8) reveals
Theenergydifferencebetweenthean“initial”
an additional term
state x at time t and a “final” state x at
t0 0 tm
θηηη˙(s)⊚(ααα(xs)−ααα¯¯¯) . time tm, may be decomposed into a telescoping
sum:
This site dependent term is clearly the local
work rate at xs. For (see equation (4)) δh|ttm0 = h(xtm,tm)−h(xt0,t0)
= h(x ,t )−h(x ,t )
θ p(xs)[ηηη˙(s)⊚(ααα(xs)−ααα¯¯¯)] tm m tm−1 m−1
xXs∈Ω +(cid:2) h(xtm−1,tm−1)−h(xtm−2,(cid:3)tm−2)
= θ p(m)[ηηη˙(s)⊚(ααα(m)−ααα¯¯¯)] +(cid:2)... (cid:3)
Xm +[h(xt1,t1)−h(xt0,t0)] . (12)
= ppp·hhh˙˙˙ . (11)
As is customary, let F be the σ−algebra gen-
t
erated by paths in Ω up to time t. For simplic-
SUMMARY ity, is supposed that F contains the informa-
0
tion about the explicit time dependence of the
Under the condition of global (site indepen- Hamiltonian.
dent) exchange rules on a simplicial complex, Also, as is customary, let E(∗|F ) denote
t
a correspondence has been shown between the the projection operator (conditional expecta-
chain groups(with realcoefficients) of the state tion) that projects its argument onto the set of
space and the thermodynamics. In the case square integrable functions on the measurable
of constant temperature, time stationary equi- space (Ω,F ) using the measure P. The accu-
t
librium, the Dynkin-L´evy kernel ties the sys- mulateddifferencebetweentheactualtrajectory
tem’slocalheatratetospecificmembersofthose andits projectionsmaybe approximatedasfol-
groups. lows
7
⊥h|ttm0 = hh(xtm,tm)−h(xtm−1,tm−1)i E(h(xtm+1,tm+1)−h(xtm,tm)|Ftm)
− E(h(x ,t )
tm m
h(x ,t )−h(x ,t )
−h(xtm−1,tm−1)|Ftm−1) = tm mt +1 −t tm m (tm+1−tm)
m+1 m
+ h(x ,t )−h(x ,t )
h tm−1 m−1 tm−2 m−2 i + [h(k,tm)−h(xtm,tm)]
− E(h(x ,t ) Xk
−h(txmt−m1−2,mt−m1−2)|Ftm−2) × qxtmk(tm) (tm+1−tm) (16)
+ ··· Under these conditions the accumulated af-
+ [h(x ,t )−h(x ,t )] fects over the the time interval [0,t] are
t1 1 t0 0
− E(h(x ,t )−h(x ,t )|F ) (13)
t1 1 t0 0 t0
t ∂h(x ,s)
Itisastandardresultthat⊥h|tm isamartin- s + [h(k,s)−h(x ,s)]
t0 Z (cid:18) ∂s s
gale with respect to the filtration Ft previously 0 Xk
described.
q
In the case that the position process is × xsk(s)(cid:19)ds (17)
Markoff,thesmalltimecontributionstotheker-
nel may be computed as Atconstantθ,intermsoftheexponentsfrom
equation (7)
t
E(h(x ,t )−h(x ,t )|F )
tm+1 m+1 tm m tm θ η˙(s)⊚[α(xs)−α¯]
Z
0
= [h(xtm,tm+1)−h(xtm,tm)] + η(s)⊚[α(k)−α(xs)]
×(1−O(t −t )) Xk
m+1 m q
× (s) ds (18)
+ [h(k,t )−h(x ,t )] xsk
m+1 tm m
k6=Xxtm At constant θ, in terms of fancier expo-
P nents resulting from the application of time
× xtmk(tm−1,tm) (14) varying microforces (local exchange rules and
more precisely: timescales)
t
θ ds η˙(s)⊚[α(x )−α¯]
E(h(xtm+1,tm+1)−h(xtm,tm)|Ftm) Z0 (cid:18) s
+ δ˙(x ,s)⊚α(x )−δ˙(∗,s)⊚α(∗)
h(x ,t )−h(x ,t ) s s
= tm m+1 tm m (t −t )
tm+1−tm m+1 m + q (s) η(s)⊚[α(k)−α(xs)]
+ [h(k,t )−h(x ,t )] Xk xsk (cid:20)
m+1 tm m
k6=Xxtm +δ(k,s)⊚α(k)−δ(xs,s)⊚α(xs)
q (cid:21)(cid:19)
× xtmk(tm) (tm+1−tm) . (15) (19)
Assuming that
h(k,s+)−h(k,s)
is small. for example h as a differentiable func-
tion of time, this is ∗ [email protected]
8
[1] D.Ford,“ApplicationofThermodynamicstothe 1981
ReductionofDataGeneratedbyaNon-Standard [6] A.J.F.Siegert, “OntheApproachtoStatistical
System,” http://www.arxiv.org/abs/cond- Equilibrium” Phys.Rev.76, 1708, 1949
mat/0402325, 2004 [7] C. Godreche and J. Luck, “Nonequilib-
[2] D. Ford, “Surfaces of Constant Tempera- rium Dynamics of the Zeta Urn Model,”
tureinTime,” http://www.arxiv.org/abs/cond- http://www.arxiv.org/abs/cond-mat/0106272,
mat/0510291, 2005 2001
[3] D. Ford, “Surfaces of Constant [8] A. Lipowski and M. Droz, “Binder Cumulants
Temperature for Glauber Dynamics,” ofanUrnModelandIsingModelAboveCritical
http://www.arxiv.org/abs/cond-mat/0601387, Dimension,” http://www.arxiv.org/abs/cond-
2006 mat/0201472, 2002
[4] J.L.Doob, “StochasticProcesses,” Wiley,New [9] W. Gordon and G. Newell, “Closed Queueing
York,1953 Systems with Exponential Servers,” Operations
[5] P.Bremaud, “PointProcessesandQueues,Mar- Research, vol. 15, no. 2, pp. 254–265, 1967.
tingale Dynamics,” Springer-Verlag, New York,