Entropy Production in Collisionless Systems. II. Arbitrary Phase-Space Occupation Numbers

0.94 MB·English

by  Eric I. Barnes

Draftversion January31,2012
PreprinttypesetusingLATEXstyleemulateapjv.5/2/11
ENTROPY PRODUCTION IN COLLISIONLESS SYSTEMS. II. ARBITRARY PHASE-SPACE OCCUPATION
NUMBERS
Eric I. Barnes
Department ofPhysics,UniversityofWisconsin—LaCrosse,LaCrosse,WI54601
Liliya L. R. Williams
MinnesotaInstitueforAstrophysics,UniversityofMinnesota,Minneapolis,MN55455
2
Draft version January 31, 2012
1
0
ABSTRACT
2
We present an analysis of two thermodynamic techniques for determining equilibria of self-
n
gravitating systems. One is the Lynden-Bell entropy maximization analysis that introduced violent
a
relaxation. Since we do not use the Stirling approximationwhich is invalid at small occupation num-
J
bers, our systems have finite mass, unlike Lynden-Bell’s isothermal spheres. (Instead of Stirling,
7
we utilize a very accurate smooth approximation for lnx!.) The second analysis extends entropy
2
production extremization to self-gravitating systems, also without the use of the Stirling approxima-

tion. In addition to the Lynden-Bell (LB) statistical family characterized by the exclusion principle
]
O in phase-space, and designed to treat collisionless systems, we also apply the two approaches to the
Maxwell-Boltzmann (MB) families, which have no exclusion principle and hence represent collisional
C
systems. We implicitly assume that all of the phase-space is equally accessible. We derive entropy
. production expressions for both families, and give the extremum conditions for entropy production.
h
p Surprisingly, our analysis indicates that extremizing entropy production rate results in systems that
- have maximum entropy, in both LB and MB statistics. In other words, both thermodynamic ap-
o proaches lead to the same equilibrium structures.
r
t Subject headings: galaxies:structure — galaxies:kinematics and dynamics
s
a
[ 1. INTRODUCTION entropy maximum. In calculating the most likely state

1 1.1. Motivation it is implicitly assumed that all states of the system are
v equally accessible.
9 Understanding how collisionless systems attain a spe- SuchanapproachwastakenbyLynden-Bell(1967)and
9 cific mechanical equilibrium state is fundamentally im- applied to self-gravitatingcollisionless systems, with the
8 portanttoastrophysics. Forexample,thecolddarkmat- hope of explaining the observed light distribution of el-
5 ter structures that exist around galaxiesare expected to lipticalgalaxies. Acollisionlesssystemcanbethoughtof
. fall into this class of system. Individual dark matter as a fluid in the 6D phase-spaceof position and velocity.
1
constituents (whatever they may be) should evolve ac- The ‘particles’ in Lynden-Bell’s analysis are parcels of
0
2 cording to a mean-field gravitational potential, free of phase-space,i.e.,parcelsofthisfluid,andsothedistribu-
1 the influence of individual encounters. tion function (DF) representing the phase-space density
: The range of mechanicalequilibria available to a colli- is defined in terms of energy per unit mass, not energy
v sionless system is defined by the Jeans equation, which per particle. The analysis resulted in a DF similar to
Xi representstheconditionthatnoportionofthesystemex- theFermi-Diraccase,butwithadifferentnormalization.
periences a net force. Unfortunately, the Jeans equation Lynden-Bell(1967)arguedthatanon-degeneratelimitis
ar admits an infinity of solutions. Even if the mass distri- appropriateforstellarsystems,andthusarrivedataDF
bution is specified, there is an infinite set of acceptable similar to that of the Maxwell-Boltzmann case (an ex-
mechanical equilibria, each involving a different velocity ponential)whichresultedintheisothermalsphererepre-
distribution. For sphericalsystems,these velocity distri- sentingthermalequilibrium. Sincetheisothermalsphere
butions differ in their anisotropy profile that quantifies has an infinite extent and mass, its emergence from the
radial versus tangential motion. However, the question entropy maximization procedure, which demanded a fi-
remains, how and/or why does any one collisionless sys- nitemasssystem,presentedacontradiction. Thisappar-
tem evolve to its particular mechanical equilibrium end- ent failure of entropy maximization was puzzling, and it
state, and what are the properties of such a state? wasoftenarguedthatsuchsystemsdonothavestatesof
maximum entropy. Some effort was made to investigate
1.2. Thermodynamic Approaches to the Problem
maximizing entropy with additional constraints beyond
Thestatisticalmechanicsdescriptionofthermodynam- mass,energy,andangularmomentum(Stiavelli & Bertin
ics provides one path to obtaining the description of the 1987; White & Narayan 1987). Other routes involving
finalrelaxedstate. Afullyrelaxedsystemisthemoststa- minimum energy states of self-gravitating systems were
tistically likely state of that system, or the one with an also developed (e.g., Aly 1994).
Recently, Madsen (1996) (based on earlier work by
[email protected] Simons 1994) has pointed out that the reason for the
[email protected]
2 Barnes & Williams
system’sinfinitemasswastheuseoftheStirlingapprox- while the volume of a macro-cell is much larger than
imation,lnn!≈nlnn−n. Incontrasttosystemsusually thatofamicro-cell,itis stillverysmallcomparedto the
treatedinstandardstatisticalmechanics,self-gravitating full extent of phase-space occupied by the system. The
systems can apparently have small phase-space occupa- number of ways of organizing the n elements into the ν
i
tion numbers n, making Stirling a poor approximation. micro-cells without co-habitation is,
Specifically, these systems have regions of phase-space
ν!
or energy-space that are nearly or completely unoccu- . (2)
pied, such that n will be small. Spatially, these regions (ν−ni)!
can correspond to the center of the potential as well as
If the elements were allowed to multiply occupy micro-
its outer edge. Using Maxwell-Boltzmann statistics and
cells,as inMB statistics, this number wouldbe givenby
the exact lnn!, Madsen (1996) has found a distribution
νni.
function from entropy maximization that is very similar
To get the total number of accessible states, the pos-
to King (1966) models. Hjorth & Williams (2010) have
sible ways to distribute the N phase elements into n
shown that the energy-space occupation function N(E) i
chunks must also be included. Lynden-Bell (1967) de-
derived using a very accurate smooth approximation to
rives,
lnn!closelyresemblesthe results ofcollisionlessN-body
N! ν!
simulations (Williams et al. 2010). Ω = × . (3)
LB
n ! (ν−n )!
i i i i
Y
1.3. Brief Review of Statistical Representations
IntheMBcase,theoQnlychangeisthatthefactorialratio
From a statistical point of view, entropy is simply a in the final product term is replaced by νni.
measurementofthenumberofstatesaccessibletoapar- The LB case disallows two phase-space elements from
ticular system. This relationship is most commonly ex- inhabitingthe samephase-spacelocation,soitexplicitly
pressed quantitatively as, takesinto accountthe incompressibility of a collisionless
fluid. Since we are primarily interested in dark matter
S =k lnΩ, (1)
B halos, LB is the natural case to consider. For complete-
ness, and for the sake of having a comparison, we also
where Ω is the number of accessible states and k is
B
treat the MB case, in the Appendix.
the Boltzmann constant which serves to give entropy
We argue that the lack of an exclusion principle in
the correct thermodynamic units. As a result, count-
the MB case is equivalent to allowing collisions between
ing procedures are key to determining specific realiza-
particles. In a collisional system, particles from distant
tionsofentropy. Lynden-Bell(1967)discusseshowthere
phase-space locations can be scattered into any other
are four counting types that lead to physically relevant
phase-spacelocation,thereby increasingthe phase-space
situations. Bose-Einstein statistics follow from counting
density at the latter location. In principle, there is no
statesforindistinguishableparticlesthatcanco-habitate
limit to how high the density can get through such scat-
in the same state. When indistinguishable particles are
terings. In practice, the phase-space density probably
not allowed to share states, Fermi-Dirac (FD) statistics
can not become very high at most locations, but it can
emerge. Classical Maxwell-Boltzmann (MB) statistics
behigherthantheoriginalfine-grainedDF.Wenotethat
result from counting states available to distinguishable
it is common to use MB to representcollisionalsystems.
particles that can share states. Completing the symme-
Forexample,Madsen(1996)arguesthatitis the correct
try, systems where distinguishable ‘particles’—actually,
statisticstouse forglobularclusterswherethe relatively
parcels of phase-space—cannot co-occupy states obey
smallnumberofstarsallowsthe clustertorelaxthrough
what has become known as Lynden-Bell (LB) statistics
two-body interactions. It then makes sense that the en-
(statistics ‘IV’ in Lynden-Bell’s originalnotation). Each
ergydistributionthattheclusterwillarriveatwillbethe
typeofstatisticswillproducedifferentrepresentationsof
sameasthatinacloudofgas,whichrelaxesthroughcol-
entropy,butwewillfocusonthetwothatdealwithclas-
lisions between molecules. In the non-degenerate limit,
sical, or distinguishable particles, namely, LB and MB.
whenthemicro-cellsareverysparselypopulatedandthe
We briefly recap the notation used in Lynden-Bell
densityofthecoarse-graineddistributionfunctionisvery
(1967) before proceeding with our discussion. The six-
dimensional position-velocity phase-space (x,v) is the dilutecomparedtothatofthefine-grainedfunction,both
LB and MB distribution functions, and hence density
usual setting for determining the statistics. Imagine
profiles, will look the same.
phase-space to be divided into a very large number of
nearly infinitesimal parcels, called micro-cells, each hav-
1.4. This Work
ing volume ̟. Each micro-cell can either be occupied
or unoccupied by one of the N phase-space elements of Inthispaperweexploretwopossibleapproachestode-
the system. These elements can be thought of as repre- riving the final equilibrium state of self-gravitating sys-
senting the fine-grained distribution function, which has tems,foreachofthe twotypesofstatistics,LBandMB.
a constant density value η. Because collisionless pro- In both, we use a very accurate, smooth approximation
cesses imply incompressibility of the fine-grained distri- for lnx! valid for arbitraryoccupation numbers x. How-
bution function, the phase elements cannot co-habitate. ever, the price we pay for this improved approximation
If phase-space is also partitioned on a coarser level so is the loss of analytic solutions.
that some number ν of micro-cells occupy a macro-cell, The first approach assumes that the final state is the
then we can discuss a coarse-grained distribution func- maximum entropy state, an assumption that was first
tion. The volume of a macro-cellis then ν̟ and the ith used in the context of self-gravitating systems in 1950’s
macro-cell contains n phase elements. We assume that (Ogorodnikov1957). The second approach, again in the
i
3
context of self-gravitating systems, was first taken by 2. SUMMARYOFRESULTSFORLARGEOCCUPATION
Barnes & Williams (2011); it posits that the final state NUMBERS
correspondstotheextremumofentropyproductionrate. Barnes & Williams (2011) have investigated entropy
We explore extremizing entropy production because production in self-gravitating systems described by MB
it has not been established beyond a doubt that real and LB and have developed expressions for the entropy
or computer simulated systems do fully relax to max- production, σ for both the MB,
imum entropy states. It is possible that their steady-
state configurations do not correspond to maximum en- σ =−kB Γ ln f +1−lnN dv, (4)
tropy. Prior work on thermal non-equilibrium systems, MB ̟η η
Z (cid:20) (cid:18) (cid:19) (cid:21)
but not in astrophysical contexts, suggests that station-
and LB cases,
arystates—likemechanicalequilibrium—occurwhenen-
tropy production is extremized (e.g., Prigogine 1961; k f
Jaynes 1980; de Groot & Mazur 1984; Grandy 2008). σLB =− B Γ ln −C dv, (5)
̟η η−f
We investigate their applicability to self-gravitating sys- Z (cid:20) (cid:18) (cid:19) (cid:21)
tems in mechanical equilibrium. where constant C in Equation 5 is (lnN − 1) +
The new aspect in the present paper is that we use a (1/N) νlnν. In these expressions, f is the coarse-
i
very accurate approximation for lnx!, unlike our previ- grained distribution function and Γ is the relaxation
ous paper that assumed the Stirling approximation. As functioPnandformstheright-handsideoftheBoltzmann
Madsen(1996) has shown, replacingStirling with an ac- equation,
curate approximation (i) results in systems with finite
∂f
total mass and energy, and (ii) significantly changes the +v·∇f +a·∇ f =Γ(f). (6)
v
structure of the systems. ∂t
In all, we present four derivations; entropy maximiza- If one were to assume, for a moment, that in the above
tion for the LB and MB statistics are covered in Sec- equationf is afine-grainedDF, thenthe righthandside
tions 3.1 and A.1. Extremizationof entropy production would be zero for collisionless systems. In other words
for LB and MB statistics are carried out in Sections 3.2 Γ = 0, which means that on fine-grained scales there is
and A.2, respectively. We develop expressions for the no change in, or production of entropy; in collisionless
relaxation functions (see § 2), and use these to better systems, entropy is fixed throughout evolution. Let us
understand the evolution of coarse-grained distribution be clear, we are not advocating for a specific process as
function. We compare our results with analogous ver- a source for Γ, like collisions in a gas. The relaxation
sions of entropy production derived using the standard function simply describes the Lagrangian time rate of
Stirling approximation in Barnes & Williams (2011). change of the distribution function. Returning to our
Figure1putsthepresentpaper(BWIIinthefigure)in case where f represents the coarse-grained DF, Equa-
context. It is aschematic summaryofthe variousstatis- tion 6 states that entropy is produced [and as we argue
tical mechanical approaches to self-gravitating systems. in Barnes & Williams (2011) it happens evenin systems
The possible ways to frame the problem appears at the that haveattained macroscopicsteady-state]because on
top of the figure; one can formulate the problem in ei- microscopicscalesthefine-grainedDFcontinuestowind
therthe regularphase-space,orthe energyspace. Below and twist, which when combined with coarse-graining,
thethickhorizontallineweshowthetwodifferentroutes gives rise to non-zero entropy change. From a different
forattainingthefinalsteady-statestate: maximizingen- starting point, Chavanis (1998) develops an expression
tropy, and extremizing entropy production. Once these for the right-hand side of Equation 6 in terms of a “dif-
choices are made one has to decide whether small oc- fusion current” that relates to correlationsbetween fluc-
cupation number regime will be important or not, and tuations in the fine-graineddistribution function. While
hencewhethertousetheStirlingapproximationforlnx!, that work details the makeup of this diffusion current,
or not. In the latter case, one must then decide whether we simply focus on the broad behavior of the relaxation
to use the discrete (“discr.”) step-like, i.e. exact version function.
oflnx!, or to approximateit with some smooth function We find extremum entropy production conditions by
(“cont.”) which remains very accurate down to small setting the variation of entropy production, δσ equal to
x. Note that HW10 and the present paper use different zero. This operation gives expressions for the relaxation
but similar approximations. There is no physical reason function. For the MB case,
to introduce the exclusion principle in the energy state-
space, hence the corresponding regions are marked as (1−lnN)ΓMB(f =η)
Γ (f)= . (7)
MB
“not relevant”. The bottom entries of some columns in ln(f/η)+1−lnN
thetablecontainnamesofpaperswherethecorrespond-
The LB relaxation function is slightly more complex,
ing options were considered. K66 in parentheses below
BWIImeansthatKing(1966)resultsarenearlyidentical −CΓ (f =η/2)
LB
to ours. M = ∞ under LB67 means that Lynden-Bell ΓLB(f)= . (8)
ln[f/(η−f)]−C
(1967) final result, the isothermal sphere, had infinite
mass. Like Lynden-Bell (1967), the Barnes & Williams (2011)
Much of the background material for this work may work assumes that the large n Stirling approximation is
be found in Barnes & Williams (2011), and we briefly validforthesystemsbeinginvestigated. Here,wewillbe
summarizethesepreviouslyobtainedresultsinSection2. deriving relations analogous to Equations 4, 5, 7, and 8,
butusingaveryaccurateapproximation,afterdiscussing
the results of entropy maximization below.
4 Barnes & Williams
3. RESULTSFORARBITRARYOCCUPATIONNUMBERS function f, the entropy becomes,
The approximation that we utilize in this work is, k νf 1 νf
B
S =S − + ln +1 +
LB LB,0
1 ln2π ν̟ η 2 η
lnx!=(x+ )ln(x+1)−x+ +λ (9) ZZ (cid:20)(cid:18) (cid:19) (cid:18) (cid:19)
0,x
2 2 νf 1 νf
ν− + ln ν− +1 +λ +
where η 2 η 0,νf/η
(x2+2x+ 287) (cid:18) (cid:19) (cid:18) (cid:19)
λ0,x =−(x2+ 25x+28183). (10) λ0,ν−νf/η dxdv. (14)
12 12 Takingthevariatio(cid:3)nofthisentropyexpressiontobezero,
In the large x limit, this reduces to the usual Stirling with constant mass and energy constraints, leads to the
approximation lnx! = xlnx − x. Our approximation following condition,
is slightly different from the one in Hjorth & Williams
(F +1/2)
(2010). Comparisons between the Stirling approxi-
ln(F +1)+ −ln(ν−F +1)−
mation, Hjorth & Williams (2010) approximation, and (F +1)
Equation 9 are shown in Figure 2. The plots illustrate
(ν−F +1/2)
the function ψ(x+1) ≡ dlnx!/dx for the three cases. +
(ν−F +1)
The HW10 and Equation 9 approximations are nearly
identical. Most importantly, these two approximations dλ0,F dλ0,ν−F
+ +µ+βǫ=0, (15)
are well-behaved at x = 0, unlike the Stirling approxi- dF dF
mation.
where F = νf/η is a scaled coarse-grained distribution
functionandµ andβ areundeterminedmultipliers asso-
3.1. Entropy Maximization
ciated with mass and energy conservation, respectively.
In this section, we follow the overall path taken in The ǫ term is the specific energy of a phase element lo-
Lynden-Bell (1967) to determine the description of en- cated at position x with velocity v, ǫ = v2/2+Φ. The
tropy for the LB statistics family. Briefly, counting ar- derivative of the λ function is,
guments for phase-space macro-celloccupation are com-
bined with the statistical definition of entropy to give dλ0,F −(F +1)
= . (16)
specific representations. Variations in entropy assuming dF (F +600/576)
constant mass and energy are then set to zero in order
After substituting for these λ derivatives and combining
to determine the necessary distribution functions.
terms we have,
If one demands that multiple phase-space elements
cannot simultaneously occupy micro-cells, the multiplic-
F +1 (F −ν/2)
ity of states (Equation 3) combined with the definition ln + −
ν−F +1 (F +1)(ν−F +1)
of entropy results in, (cid:20) (cid:21)
2F2−2Fν−(1+600/576)ν−600/288
+
S =k lnN!− lnn !+ lnν!− ln(ν−n )! , F2−Fν−(600/576)ν+(600/576)2
LB B i i
" i i i # µ+βǫ=0. (17)
X X X
(11)
The inelegant ratio terms in this expression are both
where again the summations run over the number of
symmetric about F = ν/2. The second term on the
macro-cells. WewillassumethatbothN andν aremuch
left-hand side of Equation 17 has values of −1/2 when
largerthan1,sothatStirling’sapproximationisvalidfor
F = 0, 0 when F = ν/2, and 1/2 when F = ν (the
use in the firstand third terms. However,for the n and
i
maximum value of F for the LB case). The third term
ν−n terms we will use our improved approximation,
i
onthe left-hand side ofEquation17 is a nearly constant
ln2π function with a value very close to −2 for 0≤F ≤ν.
lnn !=(n +1/2)ln(n +1)−n + +λ . (12)
i i i i 2 0,ni We have not attempted to find an analytic solution
for F. Graphical solutions of Equation 17 for a series
Note that this usage does not require either n or ν −
i of ǫ values produces the picture of f/η seen in Figure 3.
n to actually be a small value, rather this keeps the
i The overall character of the distribution function is the
accounting accurate in the event that they do become
same in the non-Stirling and Stirling versions, and both
small. If these values are always large, there will be no
functions are very similar to Fermi-Dirac distribution,
difference from the Stirling approximation. The entropy
expression now reads, exp−(µ+βǫ)
f = . (18)
FD
1+exp−(µ+βǫ)
S =S −k [(n +1/2)ln(n +1)+
LB LB,0 B i i
i ThisdistributionfunctionofEquation17canbetrans-
X
(ν−n +1/2)ln(ν−n +1)+λ + formed into a density distribution using the Poisson
i i 0,ni equation and the fact that
λ0,(ν−ni) , (13)
where S = k [N(cid:3)lnN −N +M(νlnν −ln2π)], and ρ(r)= f(r,v)dv =4π f(ǫ) 2[ǫ−Φ(r)]dǫ. (19)
LB,0 B
M is the total number of macro-cells. Z Z
p
Transforming from the discrete macro-cell occupation Note that this procedure imposes an isotropic velocity
number n to the continuous coarse-graineddistribution distribution for the system.
i
5
The density andlogarithmicdensityslope correspond- tion into Equation 21, we get a lengthy expression,
ingtothenon-StirlingfunctionarepresentedinFigures4
∂
and 5, and will be discussed further in Section 4.1. (ρs )=
LB
∂t
k (F +1/2)
− B (−v·∇F) ln(F +1)+ −
ν̟ (F +1)
"
Z
3.2. Entropy Production Extremization ln(ν−F +1)+ ν−F +1/2 + ∂λ0,F + ∂λ0,ν−F −C dv−
Just because it is possible to describe thermal equi- ν−F +1 ∂F ∂F #
libriumstatesforcollisionlessself-gravitatingsystems,it
doesnotfollowthatrealsystems(eitherphysicalorsim- kB (−a·∇ F) ln(F +1)+ (F +1/2) −
ulated) must achieve them in a Hubble time. It is pos- ν̟ v (F +1)
"
sible that real systems incompletely relax, leaving them Z
isntaate.thMeramdaselnno(1n9-e9q6u)ialilbsoriupmoi,ntbsuotultonthg-altiviendvsetraytisolonwarlyy ln(ν−F +1)+ ν−F +1/2 + ∂λ0,F + ∂λ0,ν−F −C dv−
ν−F +1 ∂F ∂F
#
evolving systems, the maximization of entropy is only
temporary. This leads us to infer that the production of
k (F +1/2)
B
entropy may be a useful quantity for discussing quasi- γ ln(F +1)+ −
ν̟ (F +1)
equilibriaofcollisionlesssystems. Withthatinmind,we Z "
nowproceedtodeveloptheconditionsrequiredforather- ν−F +1/2 ∂λ
0,F
mal non-equilibrium state to be stationary. Specifically, ln(ν−F +1)+ + +
ν−F +1 ∂F
we find an expression for entropy production in the LB
statistical family and then extremize it. (As for the pre- ∂λ0,ν−F −C dv, (22)
ceding section, ananalogousderivationfor the Maxwell- ∂F
#
BoltzmannstatisticsmaybefoundinAppendixA.2.) As
discussed in Section 1.4, existing work on thermal non- where γ =νΓ/η.
equilibrium in non-astrophysical settings suggests that The first term on the right-hand side of Equation 22
stationary states occur when entropy production is ei- can be transformed into −∇·ρs v. The acceleration-
LB
ther maximum or minimum (de Groot & Mazur 1984; dependent terms in Equation 22 all disappear under the
Grandy 2008, and references therein). assumptions that the distribution function is even in v
From Equation 14 the entropy density in the Lynden- and disappears when v =v . The final representation
max
Bell case can be written as, of the time rate of change of the entropy density is,
∂
(ρs )=
LB
∂t
ρs =−kB (F +1/2)ln(F +1)− −kB ∇· [(F +1/2)ln(F +1)−
LB ν̟ ν̟
Z " Z
(ν −F +1/2)ln(ν−F +1)+ (ν −F +1/2)ln(ν−F +1)+λ0,F +λ0,ν−F −FC]vdv−
k F +1/2
λ0,F +λ0,ν−F −FSNLkBB,0#dv, (20) ν̟B Z γ(F)"ln(F +1)+ F +1 −ln(ν−F +1)+
ν−F +1/2 + dλ0,F + dλ0,ν−F −C dv. (23)
ν−F +1 dF dF
#
where, as before, F ≡νf/η.
Againassumingthevelocityfieldtobecomposedofmean
inTakingapartialtime derivativeofEquation20results and peculiar components v =v0+vp, we draw a corre-
spondence between the terms in,
∂
(ρsLB)=−∇·ρsLBv0−
∂t
∂ k ∂F
B
(ρs )=− [ln(F +1)+ k
∂t LB ν̟ ∂t B ∇· [(F +1/2)ln(F +1)−
Z ν̟
F +1/2 ν−F +1/2 Z
F +1 −ln(ν−F +1)+ ν−F +1 + (ν −F +1/2)ln(ν−F +1)+λ0,F +λ0,ν−F −FC]vpdv−
∂λ0,F + ∂λ0,ν−F −C dv, (21) kB γ ln(F +1)+ F +1/2 −ln(ν−F +1)+
∂F ∂F ν̟ " F +1
(cid:21) Z
ν−F +1/2 + dλ0,F + dλ0,ν−F −C dv. (24)
ν−F +1 dF dF
#
where C =S /Nk is a constant.
LB,0 B
Upon substituting ∂F/∂t from the Boltzmann equa- andthoseinthecontinuousversionoftheentropydensity
6 Barnes & Williams
evolution equation, (Equation 6 in Barnes & Williams slopes, α=−dlogρ/dlogr depicted in Figures 4 and 9,
2011), respectively.
∂ We argue that the LB case accurately represents colli-
∂t(ρs)=−∇·(Σ+ρsv0)+σ. (25) sionlesssystemsbecauseco-habitationofthephase-space
elements is prohibited. However, the density profiles for
The entropy flux due to random motions Σ is given
any value of ν is very different from the two cosmologi-
bytheintegralinthesecondtermontheright-handside
calmodels,Navarro-Frenk-White(NFW)(Navarro et al.
of Equation 24. The remaining term then makes up the
1996)andN04(Navarro et al.2004),whicharefitstothe
entropy production for the system,
results of cosmological N-body simulations. In contrast
to the latter, the LB density profiles have a flat central
k F +1/2
σ =− B γ ln(F +1)+ −ln(ν−F +1)+ density slope, while at larger radii α has a steeper rise
LB
ν̟ " F +1 than even that of the Plummer profile (Schuster 1883;
Z
Plummer 1911; Binney & Tremaine 1987) which is an
ν−F +1/2 + dλ0,F + dλ0,ν−F −C dv. (26) example of a polytropic system. In Figure 5, we com-
ν−F +1 dF dF # pare the profiles to those derived in Section A.1, which
arenearlyidenticalto Kingmodels; LBprofilesarepoor
As expected, the relaxation function determines the en- matchestoKingmodels. (Theverticallinesineachpanel
tropy production rate for the system. marktheradialpositionwhereα=2.) BoththeMBand
We find that the condition for the entropy production LBdensity profilesarealsodifferentfromthosefound in
term in Equation 26 to be extremized is, Hjorth & Williams(2010),asthosemodelscanhavecen-
tral density cusps.
F +1 F +1/2
Γ (f)=Q/ ln + + We conclude that applying entropy maximization to
LB
ν−F +1 F +1
(cid:20) (cid:18) (cid:19) the LB case, coupled with a very accurate approxima-
ν−F +1/2 dλ0,F dλ0,ν−F tion for lnn! produces (isotropic) density profiles that
+ + −C , (27)
are distinct from any other first-principles function, or
ν−F +1 dF dF
(cid:21) fits to the results of N-body simulations. We will return
where the integration constant is defined by, to this point in Section 4.3.
The MB case is intended to represent collisional sys-
ν+1 2(ν+2)
Q=γ (F =ν/2) − −C , tems because the co-habitation of the phase-space ele-
LB
ν/2+1 ν+600/288 ments is allowed. The density profiles with ν =100 and
(cid:20) (cid:21)
(28) 1000 are given in Figure 9a. The r scaling distance
max
and the λ derivatives are the same as those used in go- corresponds to the radius where a non-moving particle
ing from Equation 15 to 17. This relaxation function has an energy for which the distribution function dis-
is shown in Figure 7 and is very similar to the corre- appears. The second panel of that figure (9b) shows
sponding function from Barnes & Williams (2011), their the slope of the logarithmic density profile α along with
Figure 2. curves corresponding to three other well-known analyt-
ical density profiles. These MB solutions are basically
4. SUMMARYANDDISCUSSION
identical to King models (see Figure 10). This result
In an attempt to better understand the evolution comes as no surprise in the light of Madsen (1996), who
of self-gravitating collisionless systems, we have re- did not use any approximation for lnx!, and obtained
investigated a standard statistical mechanics approach density profiles very similar to King’s. In effect, aban-
to finding equilibria (entropy maximization), and con- doning the Stirling approximation and treating the low
tinued to develop a new approach (extremization of en- occupation number limit with the respect it apparently
tropy production), first applied to self-gravitating sys- deserves, demonstrates that the King distribution func-
tems by Barnes & Williams (2011). Entropy production tionistheresultofmaximizingentropyinMBstatistics,
in non-equilibrium steady-state systems has been previ- under the conditions of fixed mass and energy.
ously investigatedandfound to be useful in severalnon- The consequence of using a verygoodsmooth approx-
astrophysical systems, and so it is interesting to ask if imation to lnx!, as done here, instead of the exact ex-
the principle is relevant in gravitationalsystems. pressionwhichgivesdiscretestep-likevalues,aswasdone
Both of our approaches use a very accurate approxi- in Madsen (1996), is that the former results in spot-on
mation for lnn!, and not the standard Stirling formula matches to the King profiles, while the latter display
valid only for large occupation numbers. It has been re- modestdifferencesfromtheKingprofileshape,especially
cently shown that using an approximation that reflects for low Ψ(0)/σ values; see Figure 1 Madsen (1996).
correctbehaviorforsmalln,andtheprincipleofentropy The value of ν appears to play a role analogous to
maximization leads to density distributions that resem- the King scaling factor Ψ(0)/σ2, where Ψ(0) is a rela-
ble globular clusters (King profiles; Madsen 1996), and tive potential energy at the center, and σ is an energy
simulated pure dark matter halos (Williams & Hjorth scaling constant (Binney & Tremaine 1987). Increasing
2010; Williams et al. 2010). Both types of systems have ν—or Ψ(0)/σ2—increases the concentration of the sys-
finite mass and energy, in compliance with the entropy
tem. Recallthatthe higherthe concentration,the closer
maximization constraints.
the King model is to an isothermal sphere.
We find that the concentrations of the King models
4.1. Results of Entropy Maximization
that most closely resemble the Maxwell-Boltzmannden-
Entropy maximization using the LB and MB statis- sity profiles increase as the value of ν increases. Recall
ticsproducesthedensityprofilesandlogarithmicprofiles
7
that ν is the number of fine-grained micro-cells that oc- where ν is the minimum value of ν that will cause a
min
cupy a coarse-grained macro-cell, so as ν increases the singularityforagivenC. Thisexpressionisvalidaslong
distribution function becomes “grainier”. At the same as ν &10. The importance of the singularity in both
min
time,increasingtheconcentrationofaKingmodelmakes cases is that it signals that the relaxation function will
it more closely resemble an isothermal sphere. Putting reachzerowhenthe coarse-graineddistributionfunction
these two observations together implies that increasing f reachesthefine-grainedvalueη,bringingtheevolution
thecoarsenessofthedistributionfunction(byincreasing of the distribution function to a halt.
the number of fine-grained cells contained in a coarse- To sum up, extremizing entropy production in the LB
grained cell) should result in densities that have more case leads to a relaxation function Γ vs. f/η shape
LB
isothermal aspects. that drives Γ , and hence the entropy production rate
LB
Finally,wenotethatbothofthefamiliesofprofilesob- σ , to zero, which in turn means that the endpoint of
Lb
tained here have flat density cores, while cosmologically evolutionhasamaximum(morecorrectly,anextremum)
simulated halos have density cusps. entropy. Inotherwords,the entropyproductionextrem-
ization procedure that we followed in Section 3.2 tell us
4.2. Results of Entropy Production Extremization that the final state of a self-gravitating collisionless sys-
tem is a state of maximum entropy. Since in Section 3.1
Some aspect of simulations may delay, or even dis- we derived such a state, using entropy maximization, it
allow, the maximizing of entropy necessary to achieve must be the same state.
thermal equilibrium. In such a frustrated case, it is pos- TheimportantpointinboththeLBandtheMBcaseis
sible that a collisionless system finds a stationary state that extremizing entropy production leads to relaxation
by extremizing its entropy production. As a more con- functions that drive a coarse-grained distribution func-
creteexampleofthissituation,imagineametalbarwith tion to behave like a fine-grained distribution function.
one end exposed to a blowtorch and the other end in This hasa bearingonthe ‘incompletenessofrelaxation’,
an ice bath. When the bar reaches a stationary state sometimes alluded to when describing stationary-state
with a time-independent, but spatially varying, temper- collisionless systems. Our results suggest that if incom-
ature distribution, entropy production in the bar is an plete relaxation does happen, it is not due to a system
extremum (de Groot & Mazur 1984, Ch. 5, Sec. 3). We reaching an entropy production extreme. Rather, it ap-
have begun to explore the possibility that the mechan- pearsthatcoarse-grainedevolutionwillproceeduntilen-
ical equilibria of simulated collisionless systems can be tropy production ceases, when Γ = 0, and so full relax-
explained as states of extreme entropy production. ation will be achieved in self-gravitating systems. As an
To this end, we have derived expressions for entropy immediate consequence we conclude that entropy maxi-
production in collisionless systems using the Lynden- mization is the correct—and more direct—procedure to
Bellstatistics,andincollisionalsystems,usingMaxwell- take to arrive at the description of steady-state self-
Boltzmann statistics. Further, we have found the form gravitating systems.
of the relaxation function required to guarantee an en- Totestthehypothesisthatforself-gravitatingsystems
tropy production extreme. Recall that the relaxation the state of entropy production extreme coincides with
function is the right-hand side of the Boltzmann equa- thestateofmaximumentropy,weareundertakingacom-
tion, Equation 6, that describes the coarse-grained dis- parison between these analytical descriptions of entropy
tribution function evolution. behavior in collisionless systems and results of N-body
As in our previous work which was valid only for and semi-analytical simulations (e.g., the extended sec-
large phase-space occupation numbers, the Maxwell- ondaryinfallmodelESIMWilliams, Babul, & Dalcanton
Boltzmann relaxation function (pictured in Figure 6) is 2004; Austin et al. 2005). This further work will help
positive for all values of F (as long as N > 1) and be- settle exactly what role entropy production plays in de-
comes more constant as the number of particles N in- termining collisionless equilibria.
creases. It is clear that the basic form of the relaxation
function is not dependent on the specifics of the distri- 4.3. Outstanding Questions
bution function. A visual comparison between Figure 6
andFigure1inBarnes & Williams(2011)reinforcesthis Aside from the future work described above, to quan-
point. tify entropy production rate in simulated numerical sys-
The general form of the Lynden-Bell relaxation func- tems, there are also a few questions that remain unan-
tion shown in Figure 7 has similarities and differences swered.
with its large occupation number version derived in
Barnes & Williams (2011). The similarities are that 4.3.1. Maximizing Entropy in Phase-space vs. Energy-space
both the functions increase with f/η up to just short of Hjorth & Williams (2010) andthe presentpaper max-
f/η=1,thenspiketowardslargepositivevaluesofΓLB, imized entropy, but in different state-spaces, energy and
gothroughasingularityatf/η<∼1,thenbecomenegative, (x,v) phase-space, respectively. The results of the two
and eventually asymptote to ΓLB = 0 as f/η → 1 (see studies, for example in terms of the density profiles,
the bottom panel of Figure 7). The difference is that are very different from each other. It is not surprising
the large occupation number ΓLB is independent of ν, that they are different, but it is not immediately obvi-
while its arbitrary occupation number analogue is not. ous which one of the two approaches would produce a
Inthelatter casethe valuesofC andν thatproducethe better description of the results of collisionless N-body
singularity are linked through the relationship, simulations. After looking at all the density profiles it is
seen that Hjorth & Williams (2010) profiles are similar
ν
min
log =0.4348(C−4.1518), (29) to those of simulated systems (Williams & Hjorth 2010;
100
8 Barnes & Williams
Williams et al. 2010), while those from the LB case in The steady-state of self-gravitating systems can ap-
the present work (Section 3.1, Figures 4 and 5) are not. parently be calculated as the maximum entropy state.
WenotethatourLBentropymaximizationresultsare Thoughthesesystemsareinmechanicalequilibrium,one
whatLynden-Bell(1967)wouldhaveobtainedhadhenot could argue that they are not in conventional thermal
usedtheStirlingapproximation. Theproceduredoesnot equilibrium because their kinetic temperature is mani-
rely on any approximations, and is well motivated from festlydifferentacrossthesystem. Canwereconcilemax-
firstprinciples,assumingthatallstatesareequallyacces- imum entropydefining thermodynamic equilibriumwith
sibleandefficientmixinginphase-spacecanbeachieved. the resulting non-constant temperature? (Note that a
This begs the question, do such systems exist? More realistic self-gravitating system cannot have a constant
workwithsimulationsisneededtoaddressthisquestion. kinetic temperature, because there is no finite-mass so-
lutiontothe Jeansequationwithconstantσ.) This con-
4.3.2. The Necessity of the Low Occupation Number Regime tradiction is part of the reason why it made sense to
Our MB entropy maximization results of Section A.1 conclude, based on Lynden-Bell (1967) work that there
(and the very similar results of Madsen 1996) de- isnomaximumentropystateforself-gravitatingsystems.
scribe some globular clusters well (Elson et al. 1987; Weproposethatforgravitationally-boundcollisionless
Meylan & Heggie1997;Williams et al.2011),andthere- systems,onemustcarefullyseparatekinetictemperature
sults of Hjorth & Williams (2010) describe N-body and TkfromthermodynamictemperatureTt. Tobeclear,the
ESIM halos. These facts lead us to the following chain kinetic temperature we are referring to is related to the
of reason. First, globular clusters, along with N-body rmsvalueofthe peculiarvelocityinasystem, Tk ∝hvp2i.
and ESIM halos, are self-gravitating systems. Second, The thermodynamic temperature is linkedto the energy
the entropy maximization procedures that produce suc- scaleβ thatservesasa Lagrangemultiplierthroughβ =
cessfulmodelsofthesesystemsrequirecorrecttreatment 1/(kBTt). Bydefinition,Ttmustbeaconstant,andthere
of low phase-space occupation numbers. Therefore, it is no contradiction as in Lynden-Bell (1967)—systems
appears that the low occupation number regime is an in the maximum entropy state are characterized by a
important distinction between self-gravitating and non- constantthermodynamictemperature,evenastheyhave
self-gravitating systems. a varying kinetic temperature.
Why is this the case? We speculate that self- We can also address a related question involving the
gravitatingsystemsarespecialnotonlybecausetheyare temperature and energy change that appear in the ther-
finite inextentdue tothe long-rangenatureofthe force, modynamicdefinitionofentropy,dS =δQ/T. Thistem-
but also because of correlations between spatial regions perature must be a constant, so we would associate this
and energies of particles. Specifically, the center of the with the thermodynamic temperature Tt. δQ (normally
potential contains phase-space elements with the most called heat) refers to all non-work exchanges of energy,
boundenergies,whiletheoutskirtsarepopulatedbyele- and so must be replaced by δǫ because in collisionless
mentswiththeleastbound,orevenzeroenergies. These self-gravitatingsystemswhatisbeingtransferredistotal
requirementsarepeculiartoself-gravitatingsystems,and energy, potential and kinetic, not just kinetic. Entropy
apparently necessitate low occupation numbers because changescanoccurduetochangesinmassdistributionas
the numbers of particles in these regions must approach well as kinetic energy transport.
zero.
However, the above hypothesis does not explain why
The authors gratefully acknowledge support
the discrete version of lnx! must be smoothed to repre-
from NASA Astrophysics Theory Program grant
sent real systems. This smoothing, though apparently
NNX07AG86G. We also thank our anonymous referee
necessary, is still not well justified.
for severalhelpful suggestions.
4.3.3. Should Maximum Entropy Imply Constant
Temperature?
APPENDIX
MAXWELL-BOLTZMANN STATISTICS
Entropy Maximization
Using the results of § 1.3, the multiplicity function for a system obeying MB statistics combined with the definition
of entropy (Equation 1) produces,
S =k lnN!− lnn !+Nlnν , (A1)
MB B i
" #
i
X
where the summation involving the macro-cell occupation n runs over the number of macro-cells. We will assume
i
that N ≫1, so that lnN!=NlnN −N. However,we will not use the Stirling approximationfor the second term on
the right-hand side of Equation A1. With our approximation, Equation 12, we can now rewrite the entropy as,
S =S −k [(n +1/2)ln(n +1)+λ ], (A2)
MB MB,0 B i i 0,ni
i
X
where S =Nk [ln(Nν)−Mln2π/2N] is a constant, and M is the total number of macro-cells.
MB,0 B
9
We now replace the discrete macro-cell occupation number n with the coarse-grained distribution function f to
i
produce
k νf νf
S =S − B +1/2 ln +1 +λ dxdv. (A3)
MB MB,0 ν̟ η η 0,νf/η
ZZ (cid:20)(cid:18) (cid:19) (cid:18) (cid:19) (cid:21)
Tomaximizethisentropyfunction, wesetδS =0,subjecttoconstantmassandenergyconstraints. Theexpression
MB
that results is,
νf (νf/η+1/2) η∂λ
0,νf/η
ln +1 + + +µ+βǫ=0, (A4)
η (νf/η+1) ν ∂f
(cid:18) (cid:19)
where µ and β are undetermined multipliers associated with mass and energy conservation, respectively. The ǫ term
is the specific energyofa phase elementlocatedatpositionx with velocityv, ǫ=v2/2+Φ. Note that ifthe standard
Stirling approximation had been employed, the second and third terms would be absent and the logarithmic term
would simply read ln(νf/η). In this case, the usual, physically inconsistent, MB distribution function would result.
As before, we make a change of variables, F ≡νf/η, and the derivative of the λ function is given by Equations 16.
Equation A4 can then be re-cast as,
(264F +276)
ln(F +1)− +µ+βǫ=0. (A5)
576(F +1)(F +600/576)
Again, we have not searched for an analytical solution, but graphical solutions for F for a range of ǫ values can be
combined to produce a plot of F(ǫ). Specifically, Figure 8 illustrates the behavior of the normalized coarse-grained
distributionfunction f/η fora particularvalue ofν. Values ofthe Lagrangemultipliers µ andβ wereadjustedso that
ǫ is always positive. Because MB statistics have no exclusion principle, f/η can have values above 1.
Entropy Production Extremization
We begin by writing entropy in terms of entropy density,
S = ρsdx, (A6)
Z
where ρ is mass density, s is the specific entropy, andthe integralis takenover the spatialextent of the system. From
the entropy form given in Equation A3, the entropy density can now be written as,
k S
ρs =− B (F +1/2)ln(F +1)+λ −F MB,0 dv, (A7)
MB 0,F
ν̟ Nk
Z (cid:20) B (cid:21)
where, as earlier, F =νf/η.
Taking a partial time derivative of Equation A7 results in,
∂ k ∂F (F +1/2) ∂λ
(ρs )=− B ln(F +1)+ + 0,F −B dv, (A8)
MB
∂t ν̟ ∂t (F +1) ∂F
Z (cid:20) (cid:21)
where B =S /Nk is a constant.
MB,0 B
Substituting ∂F/∂t from the Boltzmann equation into Equation A8 results in a lengthy expression, similar to
Equation 22 in the Lynden-Bell case. We will deal with this expression term by term. For reference, the expression
after substitution is,
∂ k (F +1/2) ∂λ
(ρs )=− B (−v·∇F) ln(F +1)+ + 0,F −B dv−
MB
∂t ν̟ (F +1) ∂F
Z (cid:20) (cid:21)
k (F +1/2) ∂λ
B (−a·∇ F) ln(F +1)+ + 0,F −B dv+
v
ν̟ (F +1) ∂F
Z (cid:20) (cid:21)
k (F +1/2) ∂λ
B γ ln(F +1)+ + 0,F −B dv, (A9)
ν̟ (F +1) ∂F
Z (cid:20) (cid:21)
where γ =(νΓ)/η.
The process to evaluate these integrals is very similar to what has been discussed in the Lynden-Bell case. Using
the fact that v·∇F =∇·Fv, the first integral on the right-hand side of Equation A9 can be re-written as,
−∇· [(F +1/2)ln(F +1)+λ −FB]vdv. (A10)
0,F
Z
Note that this expression is reminiscent of the form of the entropy density in Equation A7.
We next turn our attention to the terms involving acceleration in Equation A9. The fact that a is velocity-
independent implies that a·∇ f = ∇ ·af, a fact that will be used often in dealing with these terms. Let us start
v v
10 Barnes & Williams
with the first part of the the second term on the right-hand side of Equation A9,
(−a·∇ F)ln(F +1)dv =− (∇ ·aF)ln(F +1)dv. (A11)
v v
Z Z
Integrating by parts produces two terms, one with the form,
∇ ·[aF ln(F +1)]dv. (A12)
v
which equals zero after using the divergencetheorem and the fact that any physicaldistribution function must vanish
for large velocities. The other term resulting from the integration by parts is,
F
a· (∇ F) dv =a· ∇ [(F +1)−ln(F +1)] dv. (A13)
v v
F +1
Z Z
A single component of this integral will appear as,
v1,max ∂
[(F +1)−ln(F +1)] dv =(F +1)−ln(F +1)|v1,max, (A14)
∂v 1 v1,min
Zv1,min 1
where v =−v , representingthe maximum speed possible for the system. Assuming that F is evenin v
1,min 1,max 1,max
and that F(v ) = 0 so that the distribution function disappears at the maximum speed, this integration results
1,max
in zero. Similar manipulations can be applied to the rest of the acceleration-dependent parts of the second term on
the right-hand side of EquationA9, resulting in the entire second term being equalto zero. EquationA9 now has the
form,
∂ k
(ρs )=− B −∇· [(F +1/2)ln(F +1)+λ −FB]vdv+
MB 0,F
∂t ν̟
(cid:26) Z
F +1/2 ∂λ
γ(F) ln(F +1)+ + 0,F −B dv. (A15)
F +1 ∂F
Z (cid:20) (cid:21)(cid:27)
Assuming the velocity field to be composed of mean and peculiar components v = v0 +vp, we can re-cast Equa-
tion A15 as,
∂ k
(ρsMB)=−∇·ρsMBv0− B −∇· [(F +1/2)ln(F +1)+λ0,F −FB]vpdv+
∂t ν̟
(
Z
F +1/2 ∂λ
γ(F) ln(F +1)+ + 0,F −B dv . (A16)
F +1 ∂F
Z (cid:20) (cid:21) )
WenowequatethetermsinEquationA16tothoseinthecontinuousversionoftheentropydensityevolutionequation,
Equation 25,
∂
(ρs)=−∇·(Σ+ρsv0)+σ. (A17)
∂t
The entropy flux Σ is givenby the integralin the second term on the right-handside of EquationA16 and represents
randomly fluxed entropy. The remaining term is the entropy production for the system,
k F +1/2 ∂λ
σ =− B γ(F) ln(F +1)+ + 0,F −B dv. (A18)
MB
ν̟ F +1 ∂F
Z (cid:20) (cid:21)
Thisequationexplicitlydemonstrateshowthenon-collisionlessnatureofthecoarse-graineddistributionfunctionleads
to changes in entropy.
Asmentionedin§1,thermodynamicnon-equilibriumsystemscanhavesteady-statesdescribedbyextremaofentropy
production. We then set δσ =0. Taking the variation of Equation A18, we find
MB
k dγ F +1/2 dλ 2 F +1/2 d2λ
δσ =− B δF ln(F +1)+ + 0,F −B +γ − + 0,F dv. (A19)
MB ν̟ dF F +1 dF F +1 (F +1)2 dF2
Z (cid:26) (cid:20) (cid:21) (cid:20) (cid:21)(cid:27)
Since δF is arbitrary, the variation disappears only when the term in curly braces is zero. The condition for an
extremum in entropy production is,
dlnγ F +1/2 dλ 2 F +1/2 d2λ
0,F 0,F
ln(F +1)+ + −B + − + =0. (A20)
dF F +1 dF F +1 (F +1)2 dF2
(cid:20) (cid:21) (cid:20) (cid:21)
The solution for the Maxwell-Boltzmann relaxation function is,
F +1/2 dλ S
0,F MB,0
γ (F)=P/ ln(F +1)+ + − , (A21)
MB
F +1 dF Nk
(cid:20) B (cid:21)