A&A 558, A40 (2013) Astronomy

DOI: 10.1051/0004-6361/201322122 &


c ESO 2013 Astrophysics

Incomplete-exclusion statistical mechanics in violent relaxation

(Research Note)
R. A. Treumann1,2 and W. Baumjohann3

1
International Space Science Institute, 3012 Bern, Switzerland
e-mail: Wolfgang.Baumjohann@oeaw.ac.at
2
Department of Geophysics, Munich University, 80333 Munich, Germany
3
Space Research Institute, Austrian Academy of Sciences, 8042 Graz, Austria
Received 24 June 2013 / Accepted 27 August 2013

ABSTRACT

Violent relaxation was proposed half a century ago as responsible for non-collisional dynamics in gravitationally bound systems of
extended celestial objects after reaching an equilibrium state that can be described thermodynamically. It is based on a complete
spatial exclusion principle that formally leads to a distribution function resembling the Fermi distribution. We extend this theory to
incomplete spatial exclusion by assuming that Fermi states can only be partially occupied. This is made possible by analogy to Fermi
statistics. A new form of distribution function has been obtained. Formally it does not resemble the Fermi distribution. It consists of a
difference of two Bose distributions but has the same properties as the Fermi distribution. Using it in the violent relaxation equation for
the global gravitational potential extends the violent relaxation theory to incomplete exclusion. Though this refines violent relaxation
theory, it does not resolve its basic deficiencies: the mass problem and those problems related to the mean field assumption.
Key words. equation of state – galaxies: interactions

1. Introduction in the counting-of-states procedure. They are identified with an

inverse temperature and a chemical potential, respectively. Since
Fermi statistics got a new application when Lynden-Bell (1967) the assumption of exclusion in the counting-of-states procedure
suggested his “principle of (volume) exclusion” to arrive at a is identical to the basic assumption of Pauli’s exclusion principle
statistical mechanical description of interacting stellar systems that underlies Fermi statistics, the above result is obvious, even
and of the formation of galaxies. The underlying idea was that without derivation. The proportionality factor is defined through
two extended celestial bodies cannot occupy the same volume the mass distribution of the interacting objects.
at the same time. This culminated in so-called violent relax-
 the above distribution in Poisson’s equation ∇ ψ =
2
With
ation statistics of gravitationally bound non-collisional systems,
which presumably leads to a final “thermodynamic equilibrium” −4πG fLB d v for the gravitational potential ψ, and v the ve-
3

state. Violent relaxation theory found widespread application locity of gravitationally interacting “particles” (objects: stars,
(for a review see Gott 1977) in galaxy formation and clusters galaxies, etc.), this yields Lynden-Bell’s “equation of violent re-
of galaxies, in particular in the presence of cold dark matter laxation” of a bound non-collisional system:
(Treumann et al. 2000). It also received critical reviews (Shu ∞
1978; Arad & Lynden-Bell 2005; Chavanis 1998) and improve- ζ 2 dζ exp(−ζ 2 /2)
ments (Kull et al. 1996, 1997; Lynden-Bell 1999; Lynden-Bell ∇2 ψ = −C · (2)
exp[−β(ψ + μ)] + exp(−ζ 2 /2)
& Lynden-Bell 1999; Arad & Lynden-Bell 2005). 0
Indeed, two gravitationally interacting bodies necessarily ex-
clude each other from occupying the same volume; otherwise, The integral on the right is the mass density, a self-consistent
they collide, merge, or are disrupted – all complicated non- function of both chemical potential μ and gravitational poten-
equilibrium processes outside any simple statistical mechanical tial ψ. Integration is over the velocity space volume. The dimen-
sional constant is C = 16π2Gβ− 2 η, and η is some initial phase
3
description.
space density (for its specification see, e.g., Chavanis 1998),
the normalization constant appearing in Eq. (1). The distribu-
2. Complete exclusion in violent relaxation tion fLB – or the equivalent Eq. (7) derived below – can be taken
Within Lynden-Bell’s exclusion principle, the obtained volume- for the equilibrium distribution in the Vlasov theory of linear
excluding equilibrium distribution, eigenmodes.
Violent relaxation based on fLB is a mean-field theory (cf.,
exp{−β( − μ)} e.g., Chavanis et al. 2002, 2005; Chavanis & Ispolatov 2002;
fLB ∝ , (1) Chavanis & Bouchet 2005; Bouchet et al. 2010). It supposes that
1 + exp{−β( − μ)}
a mean gravitational potential field can be defined and that a ther-
formally resembles the Fermi distribution of zero-spin fermions. modynamic equilibrium state of self-gravitating matter moving
Here,  is energy and β and μ are the Lagrange multipliers arising in this field can be achieved.
Article published by EDP Sciences A40, page 1 of 4
 A&A 558, A40 (2013)

The distribution fLB , if applied to gravitating systems, suffers distribution had profound consequences for the statistical me-
from several deficiencies. Integration over an infinite spatial vol- chanics of solids at low temperatures. Fermi’s assumption was
ume causes the total mass to diverge. This can be circumvented justified by the Pauli principle and, for spin- 21 particles, was ulti-
by restricting it to a finite – say spherical – volume (cf., e.g., mately given its quantum mechanical interpretation based on the
Arad & Lynden-Bell 2005, and others). The most severe defi- complete asymmetry of fermionic wave functions. Discovery of
ciency is that self-gravitating systems barely reach the supposed the quantum Hall effect, in particular the fractional effect, had
stationary thermodynamic quasi-equilibrium. Violent relaxation temporarily shaken Fermi statistics, until Laughlin (1983) pro-
is only an intermediate state in their gravitational evolution. posed his wave function, which includes interaction with collec-
Cosmological expansion may balance the large-scale merging tive bosonic fields.
on the universal scale. This seems to occur in cosmological co- One may ask what happens when enough states are available
moving particle-in-cell simulations. Our improvement of the ba- for fermions, e.g., electrons, and the electrons are allowed to fill
sic distribution function below is also in the realm of mean field those states incompletely, e.g., in fractions? We may think of
theories. It resolves neither of these principal problems. the loosely interacting stars or galaxies mentioned above, but for
clarity we instead stay in the framework of having to deal with
elementary particles, fermions in our case.
3. Incomplete population of states There are no experimentally known examples of incom-
The assumption of complete volume exclusion is very strong, pletely filled fermionic states. Fermion populations are binary.
even for stars. Stars have extended atmospheres, are magnetized, However, as for a theoretical example, one may think of small
and they blow out winds. Galaxies are extended objects that by numbers of gyrating electrons that bounce in a magnetic mir-
no means resemble solid bodies or point-like particles. They oc- ror geometry at frequency ωb ωce . In this case, all Landau
cupy large volumes and necessarily spatially overlap when ap- levels (L = ωce (L + 12 ), L ∈ N) split into several bounce lev-
proaching each other. Considering them as point-like objects in els b = ωb (b + 12 ), b ∈ N, b/L < ωce /ωb . The total elec-
a statistical description is convenient but can only be justified as tron energy b,L = L + b in Landau level L is then shared by
a rather weak approximation of reality. the two kinds of oscillatory states of the electron (Treumann
Modern approaches have turned to massive numerical & Baumjohann 2013). Under these circumstances, all the elec-
particle-in-cell simulations of large numbers of gravitating parti- tron energy is only in the Landau levels at the magnetic mirror
cles. They allow their temporal evolution to be followed, but they points, while a substantial part of energy is transferred to the
do not deviate from the assumption of non-extended objects. bounce levels in the field minima. Bounce levels might become
Therefore, they all fall into the same category as Lynden-Bell’s partially (hence incompletely) filled under these circumstances
statistical mechanical approach (Lynden-Bell 1967, 1979) and during the short gyration time, even though only fermionic states
its refinements by Shu (1978) and others (Kull et al. 1996, 1997; are involved, and partial population of states may not necessar-
Chavanis 1998, 2006b; Chavanis & Sommeria 1998; Chavanis ily mean that the Pauli principle is violated when not involv-
et al. 2002; Lynden-Bell 1999; Lynden-Bell & Lynden-Bell ing bosonic interactions. The occupation may still be a fraction
1999; Arad & Lynden-Bell 2005). below one that the Pauli principle does not explicitly exclude.
Relaxing the assumption of complete exclusion would be de- Though this example may only serve as illustration, we sim-
sirable. Fortunately, with this goal in mind we can exploit the ply assume that fermionic states can sometimes – hypotheti-
analogy to Fermi statistics, thereby circumventing the necessity cally – become incompletely filled. In the following, we rewrite
of precisely including any object overlap in state-counting statis- the formalism for this case of partial (or incomplete) population,
tics. Below we show that the Pauli principle can be extended to keeping in mind that it possibly is never realized in elementary
the case of incomplete (or partial) occupation of states. particle statistical mechanics, while becoming useful for some
classical systems.
Rewriting Fermi statistics for the case of partially filled states
3.1. Discrete states is easily done, starting as usual (cf., e.g., Huang 1987; Landau &
There are three kinds of particles, bosons, fermions, and anyons. Lifschitz 1994) from the logarithm Ωi [ni ] of the Γ-phase-space
Anyons obey fractional statistics (Wilczek 1982; Haldane 1983; volume corresponding to the population numbers [ni ] of the par-
Arovas et al. 1984; Wu 1994), which is a mixture of bosonic and ticles in an ideal gas:
fermionic statistics. Haldane (1991) proposed a generalization  n
μ − i i
of the Pauli principle for anyons. Bosons are allowed to occupy Ωi = −kB T log exp · (4)
energy states to arbitrary numbers; they do not resemble gravi- n
kB T
i

tationally interacting systems, so do not need to be considered

The sum is taken over Gibbs’ distributions in all the ni states.
here.
We assume that the states can become partially occupied by
For fermions, multiparticle states are inhibited by the Pauli
fermions alone. The Pauli principle categorically excludes occu-
principle, which allows for only two occupations, empty states or
pation numbers ni > 1. As a result, partial population of states
(when neglecting the particle spin) one particle per state. Fermi’s
implies that, given the number interval [0, ] with fixed natural
ingenious truncation of the infinite sum in the partition function
number ∈ N, and j any integer, such that j ∈ [0, ], the thermo-
 dynamic potential Ωi can be written as
Zi = {exp[(μ − i )/kB T ]}ni , (3)
  j
i
μ − i
Ωi = −kB T log exp , j ∈ [0, ], (5)
by boldly assuming only binary occupations ni = [0, 1] of j=0
kB T
states i with i ∈ N, immediately led him to the proposal of the
celebrated Fermi distribution ni F = {1 +exp [−(μ−i )/kB T ]}−1 . where j = 0 and j = ell just reproduce the two permitted
Here, T is the temperature; i = p2i /2m is the kinetic energy of Fermi occupations. In the intermediate interval, the popula-
the ith particle; pi is its momentum, and m its mass. The Fermi tions follow the simple partial chain { j/ }, with fixed ≥ j.
A40, page 2 of 4
 R. A. Treumann and W. Baumjohann: Violent relaxation (RN)

The sum represents a truncated geometric progression with ra- The first term on the right looks again like a Bose distribu-
tio exp[(μ − i )/ kB T ] yielding after summation tion. It is straightforward to show by expanding that, in the
  limit kB T → 0, it becomes
exp[xi ( + 1)] − 1 μ − i
Ωi = −kB T log , xi ≡ · (6)
exp(xi ) − 1 kB T ni T → 0  1 −
kB T −−−→
1, i < μ, (12)
μ − i limT = 0
One trivially shows that this becomes
the thermodynamic
 Fermi
potential ΩiF = −kB T log 1 + exp (μ − i )/kB T for = 1. By which is the completely degenerate Fermi distribution case. At
taking the derivative −∂Ωi /∂μ, the average incompletely filled high temperatures, the second term on the right cancels the first
partial distribution ni  in the ith quantum state follows as term, and the distribution algebraically approaches zero. This
 can already be seen from Eq. (10) where one has i > μ for
1 ( + 1)e xi( + 1) e xi high T , in which case the argument of the logarithm approaches
ni  = − , ≥ 1. (7)
e xi ( + 1) − 1 e xi − 1 unity. Consequently, Ωi → 0 for a high temperature.
It is easily shown that the ordinary Fermi distribution ni F is
reproduced for = 1. The partial Fermi distribution Eq. (7) 4. Discussion and conclusions
is a bit more complicated than the (integer) Fermi distribution.
Apparently, it looks more like a Boson distribution. However,
this is an illusion that becomes clear when checking its low- and In the statistical mechanics of elementary particles, to which
high-temperature forms, which agree with those for the Fermi Fermi statistics apply, the case of an incomplete population of
distribution. For T → 0, one obtains μ =  = F , ni  → 1. states is uncommon. Elementary particles like electrons either
For T μ ∼ F , one recovers the Boltzmann distribution. can or cannot be found in a particular state. The exeptions are
Thus the fermionic property of the distribution is maintained. anyons, which are collectively mixed fermionic and bosonic
Obviously it results from the subtraction of two bosonic distri- states, obeying statistics that exhibit properties of both fermions
butions in Eq. (7): The preserved antisymmetric property of the and bosons (Wilczek 1982; Haldane 1983, 1991; Halperin 1984;
partial many-particle fermionic system is caused by subtraction. Wu 1994). They play a role in the fractional quantum Hall effect
From the partial distribution ni  as a function of en- (Wilczek 1982; Arovas et al. 1984). Even though violent relax-
ergy i (p), which in its turn is a function of momentum p, all ation resembles Fermi statistics, anyon statistics have not found
thermodynamic quantities like the equation of state can be de- an application in astrophysics yet. This is not for the reason of
rived by formally defining the appropriate moments. For the its complication, but for the obvious lack of objects where its
ideal Fermi gas, one then has for pressure P and density N, application would make physical sense. Such objects might in-
deed be found in self-gravitating systems that are prototypes of
P 1 1 quasi-fermionic systems in interaction with the bosonic gravita-
= 3 f (z), N= z∂z f (z) (8) tional field. Anyon statistics might be more appropriate to them,
kB T λT λ3T
in which case exclusive fermion statistics – whether complete or
respectively, with log z = μ/kB T , when introducing an appropri- incomplete – should be replaced by anyon statistics. On the other
ate new function hand, our partially-exclusive (or incompletely- exclusive) Fermi
statistics is of a simple enough form that is suited for application
∞ ⎛ 2 ⎞
⎜⎜⎜ 1 − z1+ e−(1+ )y ⎟⎟⎟ in violent relaxation.
1 1
4
f (z) = √ y dy log ⎝⎜
2
⎠⎟ The partial Fermi distribution Eq. (7) is a variant of the (in-
π 1 − z1/ e−y2 /
0 teger occupation) Fermi distribution, which it reproduces for
∞ = 1. The low- and high-temperature limits are identical to
3 (−1)k + 1 zk/
≈ 2 , (9) those of the Fermi distribution, as is the definition of Fermi en-
k=1
k5/2 ergy. Fermions occupying partial states at T = 0 remain degen-
erate, even for the case of continuous occupation of states below
which
 replaces the common function f5/2 (z). Here, λT = filling number one. As an interesting observation, we note that
2π2/mkB T is the thermal wavelength. As usual, N = z∂z f (z) for this type of distribution the fermionic character comes about
is the particle number per volume spanned by the thermal by subtraction of two bosonic distributions.
wavelength. After this excursion into Fermi statistics we return to the vi-
olent relaxation case. For incomplete exclusion of “order ”, we
may use the new distribution function Eq. (7), which we write in
3.2. Continuous states
the conventional form as
One may formally extend the partial-occupation case to the ex- 

treme case of a continuity of partial states. Then j/ becomes a ˜f ∝ 1 exp −β( − μ)/


continuous variable in the interval j/ ∈ [0, 1], and the sum in 1 − exp −β( − μ)/
the expression for the thermodynamic potential Ωi turns into an
integral yielding

( + 1) exp −( + 1)β( − μ)/
    −
 · (13)
kB T μ − i 1 − exp −( + 1)β( − μ)/
Ωi = −kB T log exp −1 · (10)
μ − i kB T
The tilde indicates that we are dealing here with the Lynden-Bell
For the average distribution distribution function, which now is reformulated for the incom-

 plete exclusion case. The degree of incompleteness is contained
exp (μ − i )/kB T kB T in the parameter > 1. Its value must be specified from observa-
ni  =
 − · (11) tions. When solving the violent relaxation Eq. (2), it is Eq. (13)
exp (μ − i )/kB T − 1 μ − i
A40, page 3 of 4
 A&A 558, A40 (2013)

that enters the density integral. Since f˜ consists of two terms, a thermodynamical state has been reached, this is a most inter-
the density integral also splits into two different terms: esting problem of mean field statistical theory. In the statistical
⎧   mechanics of gravitationally interacting many-body systems it
∞ ⎪
⎪ exp −ζ 2 /2
⎪
⎨ may not easily be realized. For Lynden-Bell’s complete exclu-
∇ ψ = −C̃ ζ dζ ⎪
2 2
⎪
   sion statistics the occurrence of phase transitions has been in-
⎪
⎩ exp −β(ψ + μ)/ − exp −ζ 2 /2
0 vestigated extensively in the canonical and microcanonical en-
  ⎫ semble theories (Chavanis 2002, 2006a) for the fermionic and
( + 1) exp −( + 1)ζ 2 /2 ⎪
⎪
⎪ also for the bosonic models.
⎬
−

 ⎪ (14)
exp −( + 1)β(ψ + μ)/ − exp −( + 1)ζ 2 / ⎪⎪ The results culminated in the identification of zeroth- and
⎭ first-order phase transitions, their parametric dependencies, and
the identification of microcanonical and canonical critical points.
and C̃ ≡ C/ . This equation has to be solved numerically for Phase transition theory has been developed further from the
any particular object by applying some Poisson solver. By fitting point of view of thermodynamic stability (Chavanis 2006a,
the data to the solution, one might be able to determine , the 2011a,b,c,d; Chavanis & Delfini 2010; Staniscia et al. 2011).
“degree of exclusion”, in our terminology. These expressions can Similar behavior for the phase diagram and existence of critical
easily be extended to formally include either a finite number or points is expected in the incomplete-exclusion violent relaxation
a continuity of mass shells (cf., e.g., Arad & Lynden-Bell 2005). statistics.
The exponentials in Eq. (7) then turn into either sums or integrals
over shells. As noted before, the incomplete-exclusion violent Acknowledgements. We thank A. Kull and H. Böhringer for their participation
relaxation that is based on this distribution, suffers from the mass in earlier work. RT thanks the referee for valuable suggestions.
problem and is intransitive in the sense discussed by Arad &
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