UNIVERSITA DEGLI STUDI DI PADOVA Dipartimento di ASTRONOMIA Thermodynamics of violent relaxation Dr. Bindoni Daniele 13 of MAY 2011
Outlines The Aims Introduction Violent Relaxation Mechanism Distribution Function Boltzmann Equation and Jeans Theorem Information Theory Lynden-Bell s Theory Shu s Theory Mass Segregation Kull, Treumann & Böringher s Theory Nakamura s Theory Conclusion
THE AIMS The aims of this research are related to the Clausius virial maximum theory (TCV) introduced by Secco (e.g., 2005) in order to explain the main features of early type galaxies Fundamental Plane. The main items are: To analize what is the end fate of a stellar collisionless gas after relaxation, according to the current literature, even if, at the moment, a deep analysis is available only for one single component. We will see how much is problematic also this limited treatment. To see if, at the end of the relaxation phase described by the different authors, the system components have the same velocity distribution regardless of their mass, as proved by the observation of the absence of mass segregation
INTRODUCTION QUESTION: what determines the particular configuration to which a self-gravitating system settles? It is possible that their present configuration crucially depends on the conditions that prevail at their birth and on the details of their evolution. However, in view of their apparent regularity, it is tempting to investigate whether their organization can be favoured by some fundamental physical principles like those of thermodynamics and statistical physics. We wonder therefore if the actual states of self-gravitating systems are not simply more probable than any other possible configuration, i.e. if they cannot be considered as maximum entropy states.
VIOLENT RELAXATION (VR) MECHANISM For most stellar systems the relaxation time due to close two-body encounters is larger than the Hubble time by several orders of magnitude. Therefore, close encounters are negligible and the fundamental dynamics is that of a collisionless system in which the constituent particles (stars) move under the influence of the mean potential Φ generated by all the other particles. Although the dynamics is collisionless, the fluctuations of the gravitational potential are able to provide an effective relaxation mechanism on a very short timescale (less then the free-fall time) and to redistribute energy between stars in this way: THIS PROCESS IS REFERRED AS VIOLENT RELAXATION (Lynden-Bell,1967) This collisionless relaxation is able to account for the regularity of star-dominated structures.
DISTRIBUTION FUNCTION (DF) fine-grained distribution function The most basic quantity in stellar systems is the fine-grained distribution function (DF) or phase-space density: yielding the mass dm(x, v, t) (or the number of objects) contained at time t within an infinitesimal phase-space volume d 6 μ = d 3 xd 3 v centered around any point (x, v) of the 6D phase-space of stellar motions (called the μ-space in statistical mechanics). Clearly, f 0, everywhere in phase-space. coarse-grained distribution function Furthermore, it is often convenient to introduce a coarse-grained distribution function: which gives the average of the fine-grained distribution function f in a small, but not infinitesimal volume elements 3 x 3 v around the phase-space points (x, v). Contrary to the fine-grained distribution, f, the value of the coarse-grained distribution, F, depends on the particular choice of partitioning the phase-space in which the volume elements 3 x 3 v are defined.
Total derivative Suppose M(t, p 1,..., p n ) is a function of time t and n variables p i which themselves depend on time. Then, the total time derivative of M is: For example, the total derivative of f(x(t), y(t)) is
BOLTZMANN EQUATION The basic equation governing the time evolution of the distribution function, DF, in collisionless stellar systems is: where: otherwise called Boltzmann s equation (or Vlasov s equation in plasma physics). the mass contained within any infinitesimal volume d 6 μ that travels in phase space along the orbits corresponding to the potential Φ is preserved. The steady-state expression of Boltzmann equation, that do not have an explicit dependence of, DF, on time, reads:
JEANS THEOREM A costant of motion in a given force field is any function C(x, v, t) that is constant along any stellar orbit; that is, if the position and velocity along an orbit are given by x(t) and v(t) = dx/dt : An integral of motion I(x, v) is any function only of the phase-space coordinates (x, v) that is constant along any orbit, that is: Every integral is a constant of the motion but the converse is not true. which reads:
JEANS THEOREM The condition for I to be an integral is identical with the condition for I to be a steadystate solution of the collisionless Boltzmann (or Vlasov) equation. It follows that f is necessarily a composite function of the phase space variables (x, v) through one or more of the integral functions I 1, I 2,. That is: The last result is known as Jeans theorem of stellar dynamics (Jeans, 1915). In its complete form it reads as follows: 'Any steady-state solution of the collisionless Boltzmann (or Vlasov) equation depends on the phase-space coordinates only through integrals of motion in the galactic potential, and any function of the integrals yields a steady-state solution of the collisionless Boltzmann equation'
INFORMATION THEORY & STATISTICAL MECHANICS Information theory provides a constructive criterion for setting up probability distributions on the basis of partial knowledge, and leads to a type of statistical inference which is called the maximum-entropy estimate The quantity x is able of assuming the discrete values x i (i =1, 2,, n). We do not know the corresponding probabilities p i to obtain x i. All we know is the expectation value of the function f(x): and the normalization condition: We wish to know the probabilities p i that allow us to obtain the expectation value of a generic function g(x). At first glance the problem seems insoluble because the given information is insufficient to determine the probabilities p i. We need of (n - 2) additional conditions before p i and then <g(x)> could be found (Jaynes, 1957).
INFORMATION THEORY & STATISTICAL MECHANICS Claude Shannon in 1948 proved that exist a quantity which is positive, which increases with increasing uncertainty, and is additive for independent sources of uncertainty, given by: In other words, the Shannon's theorem states that, if p i are a set of mutally exclusive probabilities, then the function Q is a unique function, which, when maximised, gives the most likely distribution of the p i for a given set of constraints. Since this is just the expression for entropy as found in statistical mechanics, it will be called the entropy of the probability distribution p i. Notice also how this definition quantifies the relation between entropy, disorder and information (Longair, 1984). This last is dened as follows: Obviously the higher is the information the lower is the disorder of the system.
INFORMATION THEORY & STATISTICAL MECHANICS It is now evident how to solve the problem: in making inferences on the basis of partial information we must use that probability distribution which has maximum entropy subject to whatever is known. To maximize Q subject to the constraints, one introduces Lagrangian multipliers λ, µ in the usual way to obtain the result: The constant λ, µ, are determined by substituting the expression of p i into the constraints. The result may be written in the form:
LYNDEN-BELL s APPROACH (LB,1967) The theory takes place in the µ phase-space in which the phase-space volume is conserved and the following restrictions hold: 1. The total number of elements of phase (see later) which have any given phase density, f (x,v, t), is the same as it was initially. 2. The total energy is conserved. 3. As a corollary of (1) no two elements of phase can overlap in phase-space so that the phase-space density would be different in the region of overlap. This last assumption is equivalent to admit an exclusion principle for the phaseelements, and the fact that the phase-elements are distinguishable leads to introduce a fourth type of statistics besides the classical one of Maxwell-Boltzmann and the two quantistic statistics, as follows:
We analize four theories which predict the end fate of a stellar collisionless gas after VR: 1. LB 2. SHU 3. KTB 4. NAKAMURA PROCEDURE: To do a partition of the µ-space To calculate the number W of all possible microscopic configurations that correspond to a given macrostate, To define a Boltzmann entropy: S = lnw for this particular macrostate. To determine a statistical equilibrium state as the most probable macrostate, i.e., the one which maximizes S under the constraints of mass and energy conservation
LB s MACROSTATE: discretized realization of the coarse-grained distribution function of the system at the time t. In other words that means the F (x,v, t ) is defined as a discrete function on the ith macrocell in this way: n: number of particles inside a phase-element
He calculates the number W ({n i }) of all possible microscopic configurations that correspond to a given macrostate, and define a Boltzmann entropy: S = lnw for this particular macrostate. He seeks to determine a statistical equilibrium state as the most probable macrostate, i.e., the one which maximizes S under the constraints imposed by all preserved quantities of the phase flow. MASS CONSERVATION: ENERGY CONSERVATION: He maximizes S by including the mass and energy constraints as Lagrange multipliers λ 1, λ 2 in the maximization process namely: Lynden-Bell s formula for the value F i of the coarse-grained distribution function within the ith macrocell at statistical equilibrium, therefore is: where the value of the phase-space density inside each moving phase-space element, is taken as constant (in Fig. 1 turns to be proportional to the darkness of phase-element).
Following the conventions of thermodynamics, we interpret λ 2 as an inverse temperature constant, λ 2 =β ~1/T and λ 1 in terms of an effective chemical potential µ = -λ 1 /β (or Fermi energy ). We thus rewrite F i in a familiar form reminiscent of Fermi-Dirac statistics At any rate, in the so-called non-degenerate limit F i tends to the form of a Maxwell-Boltzmann distribution (that is, the final state approaches the isothermal model):
A more general distribution function was derived by Lynden-Bell when the phaseelements can be grouped into J groups of distinct darkness j = 1,..., J. The final formula, reads: that is, it depends on a set of J pairs of Lagrange multipliers β j, μ j.
non-degenerate LIMIT: Because of: the result shows the correct coarse-grained distribution function to be a superposition of Maxwellian components whose velocity dispersion are inversely proportional to the phase-space density of the component at star mixture: PROBLEM: how to express the overall distribution of velocities in the galaxy by a single Maxwellian function?
SHU s CRITICISM (1978) stars are truly particles and not infinitesimals part of a continuum. In his opinion it must be possible to formulate a statistical description on the basis of a particulate description. LB exclusion principle is quantitatively important only at phase densities where twobody encounters are relevant. Since in such cases we deal with collisional particles, the conclusion is that Lynden-Bell s statistics always reduces in practice to Maxwell-Boltzmann statistics when applied to stellar systems. LB leads to a superposition of Maxwellian components whose velocity dispersion are inversely proportional to the phase space density of the component at star mixture. This difficulties may vanish in the particulate description for a collisionless stellar system as long as stars of different masses are initially well mixed in phase-space.
SHU s APPROACH Similarity and differences with Lynden-Bell approach are manifest: {n i } defines for both the macrostate but the meaning is different. in Lynden-Bell n i is the number of phase-elements, in Shu the number of particles.
The number W ({n i }) of all possible microscopic configurations in µ-space that correspond to a given macrostate {n i } is: The macrostate occupying the largest volume W (in µ-space) under the constraints of conservation of mass, M, and energy, E, is the most probable state in which the system ends up, according to classical statistical mechanics. Entropy is measured by, lnw, so maximizing entropy under conservation of M and E corresponds to varying the following quantity: with respect to n i in order to find its extremum. MASS CONSERVATION: ENERGY CONSERVATION: where Φ i is the macroscopic gravitational field, m is the mass of the single species of particle, v i the velocity corresponding to the center of macrocell i
The most probable state according to the statistical mechanics just described called by Shu the Lynden-Bell distribution for one type of particle, is: where is the energy per unit mass in cell i, and µ =α /β non-degenerate LIMIT: the Lynden-Bell distribution is simply the Maxwell-Boltzmann distribution,
Generalization to a collisionless stellar system with N j particles of mass m j MASS CONSERVATION: ENERGY CONSERVATION: Here the macrostate is defined by the collection {n ij } where i runs from 1 to I (the total number of macrocells) and j runs from 1 to J (the number of particles types, where J is much less than N, the total number of particles, J << N).
The general most probable state is: where μ j is the chemical potential of the jth particle type. non-degenerate LIMIT: the most probable distribution is a sum of Maxwellians with the same inverse temperature β : But the mean kinetic energy of one particle of type j in the isothermal fluid is then the square of the velocity dispersion has to be inversely proportional to mass: This implies a dramatic consequence: the mass segregation, that is the heaviest objects are more centrally concentrated.
Solution: mass segregation can be avoided if the initial condition is well mixed, i.e., the mass composition of each macrocell is the same: where n i is the total number of stars in the ith macrocell: It means: owing to the fixed ratio of the total number of stars with mass m j over the total number N of the particles, the number of stars of given mass m j within the i-th macrocell simply scales down proportionally to the total number of stars within the same macrocell, i.e., the relative composition of each macrocell is the same. Defining the coarse-grained mass density distribution function, F, as: we recover it as: where is the average mass of the macrocells.
We may write the coarse-grained entropy as: MASS CONSERVATION: ENERGY CONSERVATION: Maximizing S subject to the constraints, we obtain: 2 NOTE: the assumption that the mass distribution function is well mixed in μ-space translates the collisionless problem with J mass species to an equivalent N-body problem with a single mass m. In particular, the equilibrium distribution function is a single Maxwellian with a uniform velocity dispersion, the same for all the mass species. In this way the mass segregation is completely avoided.
MASS SEGREGATION Consider a volume V in a system with a sample of particles of mass M and a sample of particles of minor mass m. If there is equipartition of the energy it means that:, are the typical velocities of the particles of mass m and M, respectively. QUESTION: If: too,? Since the stars move in the field generated by the entire galaxy, if it was be choosen a sufficiently small volume, we have: as the particles M and m fill on average the same place. Moreover: i
Therefore, knowing that: we wonder if also the subsample of the Ms is in Viriual Equilibrium, i.e., Because we infer that gives: and because of the hypothesis M > m it results: so the M particles are not in Virial Equilibrium in the V volume. Moreover, due to a kinetic energy too lower in respect to the potential energy in that position, the particles of mass M, beeing not supported by T, collapse to the center giving place to the Mass Segregation.
Therefore, knowing that: we wonder if also the subsample of the Ms is in Viriual Equilibrium, i.e., Because we infer that gives: and because of the hypothesis M > m it results: so the M particles are not in Virial Equilibrium in the V volume. Moreover, due to a kinetic energy too lower in respect to the potential energy in that position, the particles of mass M, beeing not supported by T, collapse to the center giving place to the Mass Segregation.
KTB s CRITICISM (Kull, Treumann & Böringher, 1998) In order to solve the velocity dispersion problem, Shu (1978) applied to his particle approach very stringent assumption on the initial mean distribution function. KTB reexamine the statistical mechanics of violent relaxation in terms of phase-space elements of different densities. Considering phase-space elements of different volume but constant mass KTB find equal velocity dispersions linked to different mass elements.
KTB s APPROACH In order to incorporate the universal mass independence of final motion after the violent relaxation into the statistical mechanics picture the authors take as constant the phase-element mass, it means: In contrast to Lynden-Bell (1967), where the phase-elements have different mass but constant volume, the phase-elements considered here differ in volume and have constant mass.
The total number of microstates W ({n ij }) corresponding to a given set of occupation numbers {n ij } is: MASS CONSERVATION: ENERGY CONSERVATION: The most probable state is found by the standard procedure of maximizing, lnw, subject to the constraints of constant total energy and constant masses. Introducing the Lagrangian multipliers α j and β, the expression to be maximized is: Defining µ j =α j /β the most probable occupation numbers become:
The coarse-grained phase-space distribution F is defined as the sum of the J phase-space distributions : Substituting n ij, the coarse-grained phase-space distribution becomes finally non-degenerate LIMIT: CONCLUSION:The final state of the violent relaxation process is a superposition of Maxwellians characterized by a common velocity dispersion that is equivalent to an equipartition of energy per unit mass without regarding to the mass species. In fact, the Maxwellians are all characterized by the same temperature β 1. The fact that all the phase-elements have the same mass, m, allows to translate the same temperature into the same velocity dispersion:
NAKAMURA s CRITICISM In Shu s approach, the mass segregation can be avoided if the initial condition is well mixed. this initial condition may be too overrestrictive and then further investigation will be required to clarify the problem of mass segregation in this particulate approach. In KTB approach a conceptual defect is about the basis of the equal mass microcells considering that mass has more physical significance than phase-space volume. this statement is subjective. One may consider, for instance, that energy is more fundamental and may use microcells with equal energy. Then one would end up with a distribution completely different from the Gaussian distribution. In KTB approach a methodological defect is the size and shape of microcells. microcells with equal mass inevitably have different shapes and sizes, thus only limited combinations are allowed to fill the phase space without gaps.
NAKAMURA s CRITICISM In Shu s approach, the mass segregation can be avoided if the initial condition is well mixed. this initial condition may be too overrestrictive and then further investigation will be required to clarify the problem of mass segregation in this particulate approach. In KTB approach a conceptual defect is about the basis of the equal mass microcells considering that mass has more physical significance than phase-space volume. this statement is subjective. One may consider, for instance, that energy is more fundamental and may use microcells with equal energy. Then one would end up with a distribution completely different from the Gaussian distribution. In KTB approach a methodological defect is the size and shape of microcells. microcells with equal mass inevitably have different shapes and sizes, thus only limited combinations are allowed to fill the phase space without gaps.
NAKAMURA s APPROACH The μ-space is divided into a set of i small cells of equal volume μ = 3 x 3 v. The box-averaged distribution F i (t) is defined as: where f is the true distribution (the fine-grained distribution) and μ i indicates the volume integration over the ith cell. The box-averaged distribution F i represents the probability of finding a particle in the ith cell. AIM: To calculate the box-averaged distribution in the limit of t = given initial distribution at t = t 0., starting with a
He introduces the joint probability P i,ξ as a probability to find a particle in the ξth cell at t = t 0 and find the same particle in the ith cell at t = The initial and equilibrium distribution is calculated from P i,ξ as: INITIAL DISTRIBUTION: EQUILIBRIUM DISTRIBUTION: The initial state of the fine-grained distribution f (x,v, t 0 ) is assumed to be so smooth that we can regard it as a constant within a cell, i.e., in a cell.
The maximum entropy principle gives the inference that probability P i,ξ is the one that maximizes the following entropy under the constraints of energy and phase-space volume conservation. Using Lagrange s method to find the maximum, the equilibrium distribution becomes: This result means the equilibrium state is a single Gaussian distribution that is proportional to exp (-βε i ). Indeed we obtain: where the Gaussian distribution of the velocity is characterized by a unique mean square velocity dispersion β -1
CONCLUSIONS We can deduce the following answers related to our aims: 1. the real relaxation is probably more complicated in respect to the ideal treatements until now considered. In them there is an inconsistency or at best a lack of transitivity. 2. The fate of the system at the end of the violent relaxation is not definitively assigned. 3. Anyways it seems that if we take off the assumption that phase-element volumes are constant, a single Gaussian would characterized the proper DF of a collisionless gas with a mass mixture.