Thermodynamic Blocking in Self-Gravitating Systems

Abstract

Building upon a thermodynamic formalism, we show that one-dimensional self-gravitating systems in hydrostatic equilibrium with a uniform density are in maximal entropy states when submitted to adiabatic perturbations, i.e., perturbations that maintain the hydrostatic equilibrium. We also show that the same result holds for three-dimensional spherical systems when submitted to spherically symmetric adiabatic perturbations. We coin this phenomenon “thermodynamic blocking”, given its similarity to the more general “kinetic blocking”. This result underlines the importance of the thermodynamic formalism, which proves useful when kinetic equations break down.

1. Introduction

The statistical physics of self-gravitating systems is crucial to understand globular clusters, galaxies, or dark matter halos [1,2]. Therein, the number of particles, N, is a key parameter [3]. Indeed, if $N\to +\infty$, one is in the collisionless limit, i.e., particles interact only with the background mean potential they create. Following a short and violent evolution [4], such systems are shown to settle down to stationary states which may be far from thermodynamic equilibrium.

If, on the other hand, $N<+\infty$, then the system slowly evolves through stationary states to its thermodynamic equilibrium. (Three-dimensional self-gravitating systems with finite mass do not even have, stricto sensu, such a thermodynamic equilibrium. Equilibria are possible only if the system is enclosed in a finite box [5,6]). Such a relaxation is driven by Poisson fluctuations in the distribution function. Assuming that the mean system is linearly stable and integrable, this relaxation is described by the Balescu–Lenard equation [7,8]. This is the governing equation of relaxation in action space as being driven by resonant interactions between the particles. Let us point out two remarks. First, in one-dimensional systems, if the frequency profile is monotonic, the $1/N$ Balescu–Lenard equation predicts a vanishing flux in action space. This was first coined “kinetic blocking” in [9] and has since been the subject of extensive research (see, e.g., [10,11,12,13,14,15,16,17,18,19,20,21], and references therein). Second, if the matter distribution is uniform (in one or higher dimensions), then the Poisson equation (Equation (8) below) predicts harmonic oscillations for the particles, i.e., the orbital frequency is independent of the orbit. Because it makes orbital resonances ubiquitous, this latter property invalidates the Balescu–Lenard formalism. As such, this kinetic equation cannot provide any information about the long-term evolution of homogeneous self-gravitating systems. We emphasise that such homogeneous distributions are not an artificial academic setup; indeed, homogeneous distributions are typical in the central regions (cores) of self-gravitating systems (see, e.g., [22]).

In this work, we focus on this latter regime, namely homogeneous self-gravitating systems. Building upon a thermodynamic formalism, we show that, in these systems, matter flows (i.e., convective flows) and heat flows (i.e., conductive flows) do not contribute to entropy growth. Up to a particular approximation (see below), we conclude that these flows are prohibited in homogeneous self-gravitating systems. Phrased differently, if a self-gravitating system is initially homogeneous and in hydrostatic equilibrium, then it will remain so for a duration much longer than one would naively expect. As such, these homogeneous states end up being more stable and resilient against perturbations compared to other arbitrary hydrostatic equilibria. Such a phenomenon is reminiscent of kinetic blocking. On the one hand, it is more general because it works in higher dimensions as well. On the other hand, it is less stringent than kinetic blocking, as it prohibits only macroscopic flows, while kinetic blocking freezes microscopic flows (redistribution of actions) as well. Given these similarities, we coin this phenomenon “thermodynamic blocking”.

The structure of the paper is as follows. In Section 2, we focus on 1D self-gravitating systems. In particular, we introduce our thermodynamic formalism (Section 2.1), detail how to perform thermodynamic variations (Section 2.2), recover the expected thermodynamic equilibrium (Section 2.3), and finally uncover the phenomenon of thermodynamic blocking in homogeneous systems (Section 2.4). In Section 3, we detail how the same results also hold in 3D for spherical self-gravitating systems, when they are submitted to spherically symmetric perturbations. Finally, we briefly discuss our results in Section 4.

2. 1D Self-Gravitating Systems

One-dimensional self-gravitating systems are particularly useful and insightful models (see [23], for a thorough review): they may be used to model gravitational collapse during the formation of cosmological large-scale structures (see, e.g., [24,25]); to understand the intricacy of collisionless (violent) relaxation (see, e.g., [26,27,28]) and collisional (slow) relaxation (see, e.g., [29,30]); or even to describe the spontaneous thickening of galactic discs (see, e.g., [31]). In practice, the modelling of 1D self-gravitating systems is made more tractable owing to their reduced number of degrees of freedom. As a result, here, we first describe thermodynamic blocking in the 1D case. Most of the present results will also be valid in the 3D case (Section 3). We refer to [32,33,34,35] for detailed reviews on some of the key aspects of the thermodynamics of self-gravitating systems. Importantly, the following calculations (both 1D and 3D) are completely Newtonian. For a relativistic approach to the dynamics and thermodynamics of 1D self-gravitating systems, we refer to [36] and references therein.

2.1. From Kinetics to Thermodynamics

Let us start with the formalism of kinetic theory in 1D (see [1], for some generic introduction). We define $f(x,v,t)$ as the distribution function in the phase space spun by the coordinate, x, and its canonically conjugate momentum, the velocity, $v=\mathrm{d}x/\mathrm{d}t$. In practice, we normalise f so that $\int \mathrm{d}x\mathrm{d}vf(x,v)=M_{\mathrm{tot}}$, with $M_{\mathrm{tot}}$ the system’s total mass. With this convention, the probability of finding a particle within some (infinitesimal) volume $\mathrm{d}x\mathrm{d}v$ around $(x,v)$ is $f(x,v)\mathrm{d}x\mathrm{d}v/M_{\mathrm{tot}}$.

Assuming that the dynamics of the particles is driven by the Hamiltonian

$$\mathcal{H}=\frac{1}{2}{v}^2+\varphi \left(x\right),$$

(1)

with $\varphi$ being the potential, then f evolves according the collisionless Boltzmann/Vlasov Equation (1)

$${\displaystyle \frac{\mathsf{\partial }f}{\mathsf{\partial }t}}+v{\displaystyle \frac{\mathsf{\partial }f}{\mathsf{\partial }x}}-{\displaystyle \frac{\mathsf{\partial }\varphi }{\mathsf{\partial }x}}{\displaystyle \frac{\mathsf{\partial }f}{\mathsf{\partial }v}}=0,$$

(2)

where we used Hamilton’s canonical equations of motion (a non-zero collision operator on the right-hand side of Equation (2) would originate from pairwise interactions between particles).

As usual, we now introduce the density and velocity moments with

$$\begin{array}{cc}\hfill \rho (x,t)& =\int \mathrm{d}vf(x,v,t),\hfill \end{array}$$

(3a)

$$\begin{array}{cc}\hfill ⟨v^n⟩(x,t)& ={\displaystyle \frac{1}{\rho \left(x\right)}}\int \mathrm{d}v{v}^{n}f(x,v,t).\hfill \end{array}$$

(3b)

We can compute the velocity moments of Equation (2) to obtain the Jeans equations. More precisely, combining the zeroth and first moments, one gets [1]

$$\rho \left(\dot{⟨v⟩}+⟨v⟩{⟨v⟩}^{\prime }\right)=-{\left[\rho \left(⟨v^2⟩-{⟨v⟩}^2\right)\right]}^{\prime }-\rho {\varphi }^{\prime }.$$

(4)

In this expression, and for the rest of this section, we dropped the dependence with respect to x and t. In Equation (4), the dot (resp. the prime) denotes a partial derivative with respect to t (resp. x). Of course, Equation (4) is not closed. The dynamics of $⟨v⟩$ involves $⟨v^2⟩$, whose evolution equation itself involves higher-order moments, and so on.

Progress can be made by placing ourselves in the time-stationary limit. Let us therefore assume that there is neither time derivative ($\dot{⟨v⟩}=0$) nor bulk motion ($⟨v⟩=0$) (for an isolated system, one can always set $⟨v⟩=0$ by placing oneself in an appropriate inertial frame). Equation (4) then reduces to the simple form

$$p^{\prime }=\rho g,$$

(5)

where, once again, the dependence with respect to x was dropped, and we introduced the relevant quantities

$$\begin{array}{cccc}\hfill g& =-{\varphi }^{\prime }\hfill & \hfill & (\mathrm{Gravitational}\mathrm{acceleration})\hfill \end{array}$$

(6a)

$$\begin{array}{cccc}\hfill p& =\rho T\hfill & \hfill & \left(\mathrm{Pressure}\right)\hfill \end{array}$$

(6b)

$$\begin{array}{cccc}\hfill T& =⟨v^2⟩\hfill & \hfill & \left(\mathrm{Temperature}\right)\hfill \end{array}$$

(6c)

Importantly, Equation (5) (along with Equations (6b) and (6c)) is formally equivalent to the hydrostatic equation for an ideal gas [1]. This analogy makes sense here. Indeed, in the mean-field limit, the pairwise interactions between particles are neglected—just like we neglected any collision operator in the right-hand side of Equation (2). As a result, in this limit, the only contribution to the system’s internal energy stems from the particles’ disordered motion measured by the temperature. In that view, the specific internal energy, u, follows from the familiar formula of the specific kinetic energy, $\frac{1}{2}{v}^2$, i.e., one has

$$u=\frac{1}{2}T.$$

(7)

In 1D, an ideal gas is a system complying exactly with Equations (6b) and (7).

In a 1D self-gravitating system, the potential $\varphi$ follows from Poisson equation, namely

$${\varphi }^{″}=2\mathcal{G}\rho \mathrm{or}\mathrm{equivalently}{g}^{\prime }=-2\mathcal{G}\rho ,$$

(8)

where $\mathcal{G}>0$ is the gravitational constant. Although gravity is driving the evolution, this does not invalidate our assumption about the lack of collisions. Indeed, since 1D gravity is a long-range force [3], particles predominantly interact with the gravitational background they create (via Equation (8)) rather than through pairwise deflections. Because the present system is self-gravitating, we can use the Poisson equation (Equation (8)), to rewrite the hydrostatic relation from Equation (5) as

$${\displaystyle \frac{2\mathcal{G}{p}^{\prime }}{-g^{\prime }}}=-g{g}^{\prime }.$$

(9)

Using Equations (6b) and (7), it can be rephrased as

$${\left(2u{g}^{\prime }\right)}^{\prime }=g{g}^{\prime }.$$

(10)

Proceeding further, this can be integrated to give a (local) relation between the internal energy and the gravitational potential, namely

$$u={\displaystyle \frac{1}{2g^{\prime }}}\left(K+{\displaystyle \frac{g^2}{2}}\right),$$

(11)

where K is a constant. In practice, K is not guaranteed to remain constant once microscopic effects (collisions) are taken into account, i.e., when higher moments are also included. Equation (11) is an important relation. It states that for a 1D self-gravitating system in hydrostatic equilibrium, the internal energy field, $u\left(x\right)$, is a function of the local gravitational acceleration field, $g\left(x\right)$, and its local derivative, $g^{\prime }\left(x\right)$.

Following the previous arguments, we may treat a 1D self-gravitating system as an ideal gas embedded within its own self-generated gravitational field. We now push this thermodynamic viewpoint further. We assume that we may describe the instantaneous state of the system using only two moments of the distribution, namely the density, $\rho \left(x\right)$ (Equation (3a)), and the temperature, via $u\left(x\right)$ (Equation (7)). Naturally, these fields contain far less information about the system than the full phase space distribution. Fortunately, they are still sufficient to explore non-trivial phenomena.

We start with the first law of thermodynamics expressed with quantities per unit mass. It reads (see, e.g., [37])

$$\mathrm{d}s={\displaystyle \frac{1}{T}}\mathrm{d}u+{\displaystyle \frac{p}{T}}\mathrm{d}\left({\displaystyle \frac{1}{\rho }}\right)={\displaystyle \frac{1}{T}}\mathrm{d}u-{\displaystyle \frac{p}{T{\rho }^2}}\mathrm{d}\rho ,$$

(12)

where s is the specific entropy. In order to account more easily for self-gravity, we may recast Equation (12) using the Poisson equation (Equation (8)). More precisely, we have

$$\mathrm{d}s={\displaystyle \frac{1}{T}}\mathrm{d}u+{\displaystyle \frac{2\mathcal{G}p}{T{g}^{\prime 2}}}\mathrm{d}{g}^{\prime }.$$

(13)

From these differential relations, we can readily derive

$$\begin{array}{cc}\hfill {\left({\displaystyle \frac{\mathsf{\partial }s}{\mathsf{\partial }u}}\right)}_{g^{\prime }}& ={\displaystyle \frac{1}{T}},\hfill \end{array}$$

(14a)

$$\begin{array}{cc}\hfill {\left({\displaystyle \frac{\mathsf{\partial }s}{\mathsf{\partial }{g}^{\prime }}}\right)}_u& ={\displaystyle \frac{2\mathcal{G}p}{T{g}^{\prime 2}}}.\hfill \end{array}$$

(14b)

In 1D, the specific entropy of the ideal gas is generically [37]

$$s=\frac{1}{2}lnu-ln\rho .$$

(15)

Using the Poisson equation (Equation (8)), the specific entropy can be expressed as a function only of the internal energy and gravitational field, namely

$$s=\frac{1}{2}lnu-ln\left(-{\displaystyle \frac{g^{\prime }}{2\mathcal{G}}}\right).$$

(16)

One can check that injecting Equation (15) into Equation (14) recovers, exactly, both Equations (6b) and (7) that define the ideal gas.

With all this, we are now fully equipped to explore the equilibrium states of a 1D self-gravitating system. In particular, we will investigate the extrema of the system’s total entropy.

2.2. Thermodynamic Variations

The thermodynamic entropy of a self-gravitating system generically reads [37]

$$S=\int \mathrm{d}x\rho \left(x\right)s[g^{\prime }\left(x\right),u\left(x\right)],$$

(17)

where we dropped the boundaries of the integral for convenience, and the function s was introduced in Equation (16). (Clarifying its precise relation to the more general Gibbs entropy $S=\int \mathrm{d}x\mathrm{d}vfln\left[f\right]$ will be investigated in a future work.) Since the system is fully self-gravitating, it obeys the Poisson equation (Equation (8)). This allows us to replace the density, $\rho$, with the acceleration, $g^{\prime }$. As a result, the entropy then becomes a functional of the whole gravitational and energy fields, which we respectively denote with $\mathbf{g}={\left\{g\left(x\right)\right\}}_x$ and similarly $\mathbf{u}={\left\{u\left(x\right)\right\}}_x$. Equation (17) becomes

$$S[\mathbf{g},\mathbf{u}]=-{\displaystyle \frac{1}{2\mathcal{G}}}\int \mathrm{d}x{g}^{\prime }\left(x\right)s[g^{\prime }\left(x\right),u\left(x\right)].$$

(18)

Following a similar approach, we can introduce the energy functional. As detailed in Appendix A.1, for a generic 1D self-gravitating system, it reads

$$E[\mathbf{g},\mathbf{u}]=-{\displaystyle \frac{1}{2\mathcal{G}}}\int \mathrm{d}x\left[u\left(x\right)g^{\prime }\left(x\right)+\frac{1}{2}{\left[g\left(x\right)\right]}^2\right],$$

(19)

where the first term is the internal (“heat”) energy and the second is the self-gravitating binding energy. Just like the entropy (Equation (18)), the energy is a functional of the gravitational and internal energy profiles.

Now that we are armed with these two functionals, we may perform thermodynamic variations of them by applying perturbations in the fields $\mathbf{g}$ and $\mathbf{u}$. More precisely, we consider a change $\mathbf{g}\to \mathbf{g}+ϵ\Delta \mathbf{g}$ and $\mathbf{u}\to \mathbf{u}+ϵ\Delta \mathbf{u}$, with $(\Delta \mathbf{g},\Delta \mathbf{u})$ some prescribed perturbation fields, and $ϵ$ a small dimensionless parameter. Importantly, for now, we assume that the respective perturbations, $\Delta \mathbf{g}$ and $\Delta \mathbf{u}$, can be chosen independently from one another.

In the limit of small perturbations, i.e., in the limit $ϵ\ll 1$, the variations in the entropy functional are generically given by (see, e.g., [38])

$$S[\mathbf{g}+ϵ\Delta \mathbf{g},\mathbf{u}+ϵ\Delta \mathbf{u}]\simeq S+ϵ\Delta S+\frac{1}{2}{ϵ}^2{\Delta }^{2}S+\dots ,$$

(20)

where we shortened the notations on the right-hand side by writing $S=S[\mathbf{g},\mathbf{u}]$, $\Delta S=\Delta S$$[\mathbf{g},\mathbf{u};\Delta \mathbf{g},\Delta \mathbf{u}]$, and similarly for ${\Delta }^{2}S$. The same perturbative expansion also holds for the total energy. In Equation (20), the leading-order variation is generically given by

$$\Delta S=\int \mathrm{d}x\left[{\displaystyle \frac{\delta S}{\delta g\left(x\right)}}\Delta g\left(x\right)+{\displaystyle \frac{\delta S}{\delta u\left(x\right)}}\Delta u\left(x\right)\right].$$

(21)

In that expression, $\delta /\delta g\left(x\right)$ stands for the functional derivative, which satisfies the usual fundamental identities

$$\begin{array}{ccc}{\displaystyle \frac{\delta g\left(y\right)}{\delta g\left(x\right)}}={\delta }_{\mathrm{D}}(x-y),\hfill & \hfill & {\displaystyle \frac{\delta g\left(y\right)}{\delta u\left(x\right)}}=0,\hfill \\ {\displaystyle \frac{\delta u\left(y\right)}{\delta g\left(x\right)}}=0,\hfill & \hfill & {\displaystyle \frac{\delta u\left(y\right)}{\delta u\left(x\right)}}={\delta }_{\mathrm{D}}(x-y).\hfill \end{array}$$

(22)

where ${\delta }_{\mathrm{D}}$ is the Dirac delta. Similar expressions also hold for the energy functional, $E=E[\mathbf{g},\mathbf{u}]$.

Physically, from Equation (20), equilibria correspond to extrema of the entropy, i.e., states $(\mathbf{g},\mathbf{u})$ for which $\Delta S=0$, whatever the imposed perturbations $(\Delta \mathbf{g},\Delta \mathbf{u})$. Such equilibria are then thermodynamically stable if they correspond to maxima of the entropy, i.e., if one has ${\Delta }^{2}S<0$, whatever $(\Delta \mathbf{g},\Delta \mathbf{u})$.

We are now set to use the present formalism in two regimes. First, we will recover the well-known results that systems in hydrostatic equilibrium with a uniform temperature profile are entropy stationary points (Section 2.3). This is no surprise since these are known to be thermodynamic equilibria. Then, we will proceed with our main result: we will show that 1D homogeneous systems in hydrostatic equilibrium, albeit they are not thermodynamic equilibria, are also entropy maxima, when submitted to perturbations that preserve the hydrostatic equilibrium (Section 2.4). This is what we refer to as thermodynamic blocking.

2.3. Thermodynamic Equilibrium

We consider a generic 1D self-gravitating system, characterised by some profiles $(\mathbf{g},\mathbf{u})$. Our goal is to characterise the properties of this system’s entropy variations. Following Equation (21), to compute $\Delta S$, we need to compute two functional derivatives. As detailed in Appendix B, we find that the leading variation of the entropy is given by

$$\Delta S={\displaystyle \frac{1}{2\mathcal{G}}}\int \mathrm{d}x\left[-{\displaystyle \frac{g^{\prime }\left(x\right)}{2u\left(x\right)}}\Delta u\left(x\right)+\left\{{\displaystyle \frac{u^{\prime }\left(x\right)}{2u\left(x\right)}}-{\displaystyle \frac{g^{\prime \prime }\left(x\right)}{g^{\prime }\left(x\right)}}\right\}\Delta g\left(x\right)\right].$$

(23)

At this stage, we note that the vanishing of $\Delta S$ at thermodynamical equilibrium is not yet obvious. To make progress, we may perform a similar calculation for $\Delta E$. As detailed in Appendix B, we start from Equation (19) to find

$$\Delta E={\displaystyle \frac{1}{2\mathcal{G}}}\int \mathrm{d}x\left[-g^{\prime }\left(x\right)\Delta u\left(x\right)+[u^{\prime }\left(x\right)-g\left(x\right)]\Delta g\left(x\right)\right].$$

(24)

Let us now place ourselves at thermodynamical equilibrium. As such, we consider states $(\mathbf{g},\mathbf{u})$ such that

$$\left\{\begin{array}{cc}{\displaystyle u=u_0=\mathrm{cst}}\hfill & (\mathrm{Constant}\mathrm{internal}\mathrm{energy}\mathrm{profile}),\hfill \\ {\displaystyle {\left(2u{g}^{\prime }\right)}^{\prime }=g{g}^{\prime }}\hfill & (\mathrm{Hydrostatic}\mathrm{equilibrium}-\mathrm{Equation}\left(10\right)).\hfill \end{array}\right.$$

(25)

For the variation in the energy, we use $u^{\prime }=0$ in Equation (24), and find

$$\Delta E{|}_{\mathrm{T}}=-{\displaystyle \frac{1}{2\mathcal{G}}}\int \mathrm{d}x\left[g^{\prime }\left(x\right)\Delta u\left(x\right)+g\left(x\right)\Delta g\left(x\right)\right],$$

(26)

where the subscript “${|}_{\mathrm{T}}$” emphasises that we are computing variations around a thermodynamical equilibrium. For the variation of the entropy, using Equation (25), we may impose $2u_0{g}^{″}=g{g}^{\prime }$ in Equation (23). We obtain

$$\Delta S{|}_{\mathrm{T}}=-{\displaystyle \frac{1}{4\mathcal{G}{u}_0}}\int \mathrm{d}x\left[g^{\prime }\left(x\right)\Delta u\left(x\right)+g\left(x\right)\Delta g\left(x\right)\right].$$

(27)

Now, comparing Equations (26) and (27), we find that, at thermodynamical equilibrium, the first-order variations in the energy and entropy are simply linked via (Using Equation (7), this formula translates into the well-known $\mathrm{d}E=T\mathrm{d}S$.)

$$\Delta S{|}_{\mathrm{T}}={\displaystyle \frac{1}{2u_0}}\Delta E{|}_{\mathrm{T}}.$$

(28)

In practice, here we are focusing on internally generated perturbations. Phrased differently, the only variations $(\Delta \mathbf{g},\Delta \mathbf{u})$ that are physically allowed are the ones that conserve the total energy. As a consequence, one must have $\Delta E{|}_{\mathrm{T}}=0$. From Equation (28), we immediately conclude that, at thermodynamical equilibrium, one has

$$\Delta S{|}_{\mathrm{T}}=0.$$

(29)

This is a reassuring result. It shows that, within the present fluid formalism, thermodynamical equilibria are, indeed, stationary points. Determining whether this extremum is a minimum, maximum or an inflection point is much more challenging. Since this is not directly related to the main result of the present work paper, we do not discuss it here and refer the interested reader to Appendix B in [6].

Finally, as a closing remark, we note that the present calculation does not guarantee that the thermodynamic equilibria physically exist. Fortunately, the 1D thermodynamical equilibria genuinely exist (see, e.g., [23]), contrary to 3D [5].

2.4. Thermodynamic Blocking

We now delve into the main result of this work, namely the existence in 1D self-gravitating systems of another way in which entropy may set an extremum. We note that in any self-gravitating system, the finite number of particles, N, unavoidably sets some (weak) level of collisionality in the system. As such, Poisson fluctuations always tend to drive relaxation (being so, the perturbations $\Delta \mathbf{g}$ and $\Delta \mathbf{u}$ are typically expected to scale like $N^{-1/2}$). In practice, since we are interested in systems comprising $N\gg 1$ constituents, their long-term self-induced relaxation is slow. As such, one can assume that the systems evolve through a series of hydrostatic equilibria, i.e., through a series of states, $(\mathbf{g},\mathbf{u})$, that instantaneously comply with Equation (11). We call this evolution adiabatic. In the present context, adiabatic refers to slow processes (like, for example, adiabatic invariants in mechanics), and not to processes with zero entropy change. Because the systems are taken to always be in hydrostatic equilibrium, during their long-term evolution, their instantaneous states can be described solely with $\mathbf{g}$, leveraging the constraint from Equation (11). Along the same line of thought, physically allowed fluctuations, because they must comply with the hydrostatic equilibrium, can be fully characterised by $\Delta \mathbf{g}$.

Let us now write the entropy functional for such a system that evolves only along hydrostatic equilibria. Injecting Equation (11) into Equation (16), we can rewrite the specific entropy as a sole function of the local gravitational fields (and its derivatives). It reads

$$s_{\mathrm{A}}=\frac{1}{2}ln\left(-K-\frac{1}{2}{g}^2\right)-\frac{3}{2}ln(-g^{\prime }),$$

(30)

where special care must be taken with the signs in the arguments of the ln—since $g^{\prime }<0$ owing to the Poisson equation (Equation (8))—and we dropped a constant term for convenience (these omitted terms give zero contribution to the entropy change due to mass conservation). In Equation (30), the subscript “$\mathrm{A}$” emphasises that we are considering systems submitted to adiabatic perturbations, i.e., perturbations that maintain the hydrostatic equilibrium.

Following Equation (18), the total entropy becomes now a sole function of the gravitational field, $\mathbf{g}$. It reads

$$S_{\mathrm{A}}\left[\mathbf{g}\right]=-{\displaystyle \frac{1}{2\mathcal{G}}}\int \mathrm{d}x{g}^{\prime }\left(x\right)s_{\mathrm{A}}[g\left(x\right),g^{\prime }\left(x\right)].$$

(31)

Similarly, injecting Equation (11) into Equation (19), we can rewrite the total energy as a functional of only the gravitational field, $\mathbf{g}$. It reads

$$E_{\mathrm{A}}\left[\mathbf{g}\right]=-{\displaystyle \frac{3}{8\mathcal{G}}}\int \mathrm{d}x{\left[g\left(x\right)\right]}^2,$$

(32)

where we dropped the constant term in K.

When submitted to some perturbation $\Delta \mathbf{g}$, similarly to Equation (20), the variations in the entropy are given by

$$S_{\mathrm{A}}[\mathbf{g}+ϵ\Delta \mathbf{g}]\simeq S_{\mathrm{A}}+ϵ\Delta S_{\mathrm{A}}+\frac{1}{2}{ϵ}^2{\Delta }^2{S}_{\mathrm{A}}+\dots ,$$

(33)

with the shortened notations, $S_{\mathrm{A}}=S_{\mathrm{A}}\left[\mathbf{g}\right]$, $\Delta S_{\mathrm{A}}=\Delta S_{\mathrm{A}}[\mathbf{g};\Delta \mathbf{g}]$, and similarly for ${\Delta }^{2}S$. Following Equation (21), the variations in the entropy under adiabatic perturbations read

$$\begin{array}{cc}\hfill \Delta S_{\mathrm{A}}& =\int \mathrm{d}x{\displaystyle \frac{\delta S_{\mathrm{A}}}{\delta g\left(x\right)}}\Delta g\left(x\right),\hfill \end{array}$$

(34a)

$$\begin{array}{cc}\hfill {\Delta }^2{S}_{\mathrm{A}}& =\int \mathrm{d}x\mathrm{d}y{\displaystyle \frac{{\delta }^2{S}_{\mathrm{A}}}{\delta g\left(x\right)\delta g\left(y\right)}}\Delta g\left(x\right)\Delta g\left(y\right).\hfill \end{array}$$

(34b)

Naturally, similar relations also hold for the variations in the total energy. We are now set to compute explicitly the first- and second-order variations in the present “adiabatic” entropy and energy.

These calculations are detailed in Appendix C. We find that the variations in the total entropy are given by

$$\begin{array}{cc}\hfill \Delta S_{\mathrm{A}}& =-{\displaystyle \frac{3}{4\mathcal{G}}}\int \mathrm{d}x{\displaystyle \frac{g^{″}\left(x\right)}{g^{\prime }\left(x\right)}}\Delta g\left(x\right),\hfill \end{array}$$

(35a)

$$\begin{array}{cc}\hfill {\Delta }^2{S}_{\mathrm{A}}& ={\displaystyle \frac{3}{4\mathcal{G}}}\int \mathrm{d}x{\displaystyle \frac{{[\Delta g^{\prime }\left(x\right)]}^2}{g^{\prime }\left(x\right)}},\hfill \end{array}$$

(35b)

while the first-order variation of the total energy reads

$$\Delta E_{\mathrm{A}}=-{\displaystyle \frac{3}{4\mathcal{G}}}\int \mathrm{d}xg\left(x\right)\Delta g\left(x\right).$$

(36)

As a first sanity check, let us consider the case of a thermodynamical equilibrium undergoing adiabatic perturbations. Imposing the constraints from Equation (25), we find from Equations (35a) and (36) that, at first order, the variations in the total entropy and total energy are connected via

$$\Delta S_{\mathrm{A}}{|}_{\mathrm{T}}={\displaystyle \frac{1}{2u_0}}\Delta E_{\mathrm{A}}{|}_{\mathrm{T}}.$$

(37)

Exactly as in Equation (29), energy conservation leads to

$$\Delta S_{\mathrm{A}}{|}_{\mathrm{T}}=0.$$

(38)

This result confirms the validity of thermodynamic formalism in the adiabatic regime, too.

Now comes the main result of our present work. Indeed, the crucial point is to note that these are not the only entropy extrema under such perturbations. Indeed, let us now consider 1D homogeneous self-gravitating systems in hydrostatic equilibrium. As such, we consider systems characterised by some $\mathbf{g}$ such that

$$\left\{\begin{array}{cc}{\displaystyle \rho =\mathrm{cst}}\hfill & (\mathrm{Homogeneous}\mathrm{density}),\hfill \\ {\displaystyle {\left(2u{g}^{\prime }\right)}^{\prime }=g{g}^{\prime }}\hfill & (\mathrm{Hydrostatic}\mathrm{equilibrium}-\mathrm{Equation}\left(10\right)).\hfill \end{array}\right.$$

(39)

Because such systems are homogeneous, one has ${\rho }^{\prime }=0$, which when plugged in the Poisson equation (Equation (8)) gives $g^{″}=0$. As a result, we find from Equation (35a) that the first-order variation of the entropy vanishes for homogeneous states. Given that such systems also comply with Poisson equation (Equation (8)), we can generically rewrite the second-order variation from Equation (35b) as

$${\Delta }^2{S}_{\mathrm{A}}=-{\displaystyle \frac{3}{8{\mathcal{G}}^2}}\int \mathrm{d}x{\displaystyle \frac{{[\Delta g^{\prime }\left(x\right)]}^2}{\rho \left(x\right)}}.$$

(40)

This is always negative, since $\rho \left(x\right)>0$. As a result, we therefore have

$$\Delta S_{\mathrm{A}}{|}_{\mathrm{H}}=0;{\Delta }^2{S}_{\mathrm{A}}{|}_{\mathrm{H}}<0,$$

(41)

where the subscript “${|}_{\mathrm{H}}$” emphasizes that we are computing variations around an homogeneous hydrostatic equilibrium. In Appendix D, we provide an alternative derivation of the thermodynamic blocking, by leveraging the conservation of the barycentre.

This is a main result of this paper. It states that homogeneous 1D systems, in hydrostatic equilibrium, are entropy maxima with respect to adiabatic perturbations, i.e., perturbations that, themselves, maintain the hydrostatic equilibrium. This is a non-trivial result because homogeneous distributions, as in Equation (10), are not global thermodynamical equilibria, since their profile of internal energy is not constant. Phrased differently, in a 1D homogeneous system, heat does not flow from a hotter region to a colder one (under the assumptions we made). This is what we refer to as a “thermodynamic blocking”.

We point out that Equation (35b) applies to any arbitrary adiabatic perturbations. More precisely, it states that any hydrostatic entropy extremum (as imposed by Equation (35a)) is always a maximum in the adiabatic regime, i.e., one always has ${\Delta }^2{S}_{\mathrm{A}}<0$ (as imposed by Equation (35b)). This indicates a clear limitation of the present formalism. Indeed, an homogeneous system must ultimately evolve away from this hydrostatic entropy maximum toward the genuine thermodynamical equilibrium, i.e., the ultimate equilibrium distribution predicted by the second law of thermodynamics. Such a long-term relaxation can only be driven by higher-order microscopic effects, which are missing from the present thermodynamic formalism. We expect that such higher-order effects are accounted for by higher moments of the phase space distribution function, starting from Equation (2).

Typically, the relaxation time for a generic self-gravitating system scales like $N{T}_{\mathrm{dyn}}$, where $T_{\mathrm{dyn}}$ is the system’s dynamical time (see, e.g., [8]). In the presence of a thermodynamic blocking, as argued in this work, we expect for this dominant source of relaxation, driven by two-body effects, to vanish identically. Heuristically, we may therefore anticipate for the initial relaxation of an homogeneous self-gravitating system to occur on a (much) slower timescale of order $N^2{T}_{\mathrm{dyn}}$. This evolution toward genuine thermodynamic equilibrium could therefore be driven by three-body effects (see, e.g., [39]), whose (microscopic) contributions likely fall beyond the reach of the present thermodynamic formalism. This will be the topic of a future numerical exploration.

Finally, we point to [40,41] for another illustration of the thermodynamic formalism, therein to recover the main properties of the Jeans instability in self-gravitating systems.

3. 3D Self-Gravitating Spheres

We now apply all the previous ideas to 3D self-gravitating spheres. Given their astrophysical importance, homogeneous 3D spheres have been extensively studied, both numerically and analytically. In particular, ref. [22] showed how the usual formula for dynamical friction [42,43] fails in homogeneous spheres. This leads to super-dynamical friction [44], along with core stalling [45] and dynamical buoyancy [46]. This was also visited numerically in [47], which showed that, indeed, the self-consistent relaxation of a homogeneous sphere is significantly delayed compared to other non-uniform density profiles. Astrophysically, all these investigations are important in the context of the problem of “core stalling”, i.e., the (slow) dynamical infall of globular clusters toward the cores of dwarf galaxies (see, e.g., [48,49,50], and references therein).

In the present section, building upon the previous thermodynamic formalism, we show that homogeneous self-gravitating spheres in hydrostatic equilibrium are also thermodynamically blocked when submitted to spherically symmetric adiabatic perturbations, i.e., perturbations that maintain the hydrostatic equilibrium. Given that the mathematical details are essentially the same in 1D and 3D, we will restrict ourselves to the main physical results in the coming sections.

3.1. Thermodynamic Variations

Following the same arguments as in Section 2.1, one can readily recast the kinetic equations of a 3D system into a thermodynamic formalism. We emphasise that we assume stationarity hereafter. More precisely, in 3D, the ideal gas laws read [37]

$$\begin{array}{c}\hfill u=\frac{3}{2}T,\end{array}$$

(42a)

$$\begin{array}{c}\hfill p=\rho T,\end{array}$$

(42b)

where, to shorten the notations, we dropped the dependence with respect to the position, $\mathbf{r}$, and the time, t. And, similarly to Equation (5), a system will be in hydrostatic equilibrium if one has

$$p^{\prime }=\rho g.$$

(43)

Following the same approach as in Equation (15), the specific entropy for the 3D ideal gas is generically

$$s=\frac{3}{2}lnu-ln\rho .$$

(44)

Using the expression from Equation (44), one can check that computing the differentials in Equation (12) recovers, exactly, the ideal gas laws in 3D, as in Equation (42). In 3D, self-gravitating systems can have a negative heat capacity, which leads to the so-called “gravo-thermal catastrophe” [51].

In practice, we assume that the system is spherically symmetric, and introduce its enclosed mass profile, $m\left(r\right)$, where $r=\left|\mathbf{r}\right|$ is the radius. The enclosed mass is connected to the density via

$$\rho \left(r\right)={\displaystyle \frac{m^{\prime }\left(r\right)}{4\pi r^2}}.$$

(45)

In addition, because of spherical symmetry, Newton’s shell theorem connects the gravitational acceleration to the enclosed mass via (note that we denoted the gravitational constant with the same symbol ($\mathcal{G}$) in both 1D and 3D)

$$g\left(r\right)=-{\displaystyle \frac{\mathcal{G}m\left(r\right)}{r^2}}.$$

(46)

Following the same approach as in Equation (17), the total entropy is generically

$$S=\int \mathrm{d}\mathbf{r}\rho (\mathbf{r})s[u(\mathbf{r}),\rho (\mathbf{r})].$$

(47)

At this stage, we recall that 3D systems have well-defined equilibria only if they are enclosed in a finite volume (this is not a sufficient condition: temperature must also be above a certain threshold; see [1] for further details). As a consequence, the integrals here and below are meant between 0 and some finite radius. Leveraging spherical symmetry and Equation (45), we can rewrite Equation (47) as

$$S[\mathbf{u},\mathbf{m}]=\int \mathrm{d}r{m}^{\prime }\left(r\right)\left[\frac{3}{2}ln\left[u\left(r\right)\right]-ln\left[\rho \left(r\right)\right]\right].$$

(48)

Here, we view the total entropy as a functional of both the internal energy and enclosed mass profiles, u and m. In particular, following Equation (45), we have ρ(r) = ρ[m] (r). Following a similar approach, we may introduce the energy functional. As detailed in Appendix A.2, for a 3D self-gravitating system, it can be expressed as a function of (u, m) via

$$E[\mathbf{u},\mathbf{m}]=\int \mathrm{d}r\left[m^{\prime }\left(r\right)u\left(r\right)-{\displaystyle \frac{\mathcal{G}m\left(r\right)m^{\prime }\left(r\right)}{r}}\right].$$

(49)

Just like we did in 1D, let us now assume that the system is submitted to some perturbations $\mathbf{u}\to \mathbf{u}+ϵ\Delta \mathbf{u}$ and $\mathbf{m}\to \mathbf{m}+ϵ\Delta \mathbf{m}$. Importantly, we assume that these perturbations are spherically symmetric, i.e., Δu(r)= Δu(r) and similarly for Δm. We stress that this is a particularly stringent assumption, which makes the upcoming calculation tractable. Naturally, it would be interesting to go beyond this assumption and consider perturbations with arbitrary symmetries. This will be the topic of a future work. Following Equation (21), the first-order variation of the total entropy under such generic spherically symmetric perturbations reads

$$\Delta S=\int \mathrm{d}r\left[{\displaystyle \frac{\delta S}{\delta u\left(r\right)}}\Delta u\left(r\right)+{\displaystyle \frac{\delta S}{\delta m\left(r\right)}}\Delta m\left(r\right)\right].$$

(50)

Naturally, analogous expressions hold for the energy variation.

3.2. Thermodynamic Equilibrium

We now follow the same approach as in Section 2.3 and consider a generic 3D self-gravitating system, characterised by some profiles $(\mathbf{u},\mathbf{m})$. Our goal is to characterise the properties of this system’s entropy variations. Following Equation (50), to compute $\Delta S$, we need to compute two functional derivatives. As detailed in Appendix E, the leading variation of the entropy functional is given by

$$\Delta S=\int \mathrm{d}r\left[\frac{3}{2}{\displaystyle \frac{m^{\prime }\left(r\right)}{u\left(r\right)}}\Delta u\left(r\right)+\left({\displaystyle \frac{{\rho }^{\prime }\left(r\right)}{\rho \left(r\right)}}-\frac{3}{2}{\displaystyle \frac{u^{\prime }\left(r\right)}{u\left(r\right)}}\right)\Delta m\left(r\right)\right].$$

(51)

Similarly to the 1D case, it proves fruitful to compute the first-order variation of the total energy. As detailed in Appendix E, it reads

$$\Delta E=\int \mathrm{d}r\left[m^{\prime }\left(r\right)\Delta u\left(r\right)-\left(u^{\prime }\left(r\right)+{\displaystyle \frac{\mathcal{G}m\left(r\right)}{r^2}}\right)\Delta m\left(r\right)\right].$$

(52)

We now place ourselves at thermodynamical equilibrium, i.e., we consider states $(\mathbf{u},\mathbf{m})$ such that

$$\left\{\begin{array}{cc}{\displaystyle u=u_0=\mathrm{cst}}\hfill & (\mathrm{Constant}\mathrm{internal}\mathrm{energy}\mathrm{profile}),\hfill \\ {\displaystyle p^{\prime }=\rho g}\hfill & (\mathrm{Hydrostatic}\mathrm{equilibrium}-\mathrm{Equation}\left(43\right)).\hfill \end{array}\right.$$

(53)

Using Equation (42) in conjunction with Equation (43), we find that at thermodynamical equilibrium, one has

$$p={\displaystyle \frac{2}{3u_0}}\rho .$$

(54a)

$${\displaystyle \frac{p^{\prime }}{p}}={\displaystyle \frac{{\rho }^{\prime }}{\rho }}={\displaystyle \frac{3}{2u_0}}g.$$

(54b)

Using these relations, we can rewrite the entropy variation from Equation (51) as

$$\Delta S{|}_{\mathrm{T}}={\displaystyle \frac{3}{2u_0}}\int \mathrm{d}r\left[m^{\prime }\left(r\right)\Delta u\left(r\right)+g\left(r\right)\Delta m\left(r\right)\right].$$

(55)

Similarly, we can rewrite the energy variation from Equation (52) as

$$\Delta E{|}_{\mathrm{T}}=\int \mathrm{d}r\left[m^{\prime }\left(r\right)\Delta u\left(r\right)+g\left(r\right)\Delta m\left(r\right)\right],$$

(56)

where we used the relations from Equations (45) and (46). Now, comparing Equations (55) and (56), we find therefore that, at thermodynamical equilibrium, the first-order variations in the entropy and energy are simply linked via

$$\Delta S{|}_{\mathrm{T}}={\displaystyle \frac{3}{2u_0}}\Delta E{|}_{\mathrm{T}}.$$

(57)

Naturally, this bears similarities to the relation from Equation (28) obtained in the 1D case. Since we are focusing on internally generated (spherically symmetric) perturbations, $(\Delta \mathbf{u},\Delta \mathbf{m})$, we must have ${\Delta E|}_{\mathrm{T}}=0$ up to first-order accuracy, to comply with energy conservation. As a consequence, from Equation (57), we find that, at first order, at thermodynamical equilibrium, one has

$$\Delta S{|}_{\mathrm{T}}=0.$$

(58)

This is good news. We recover that, in 3D, the thermodynamical equilibria are indeed entropy extrema. We emphasise, however, that in deriving this result we limited ourselves to spherically symmetric perturbations only. A more general study is left for future work.

Analogously to the 1D case in Section 2.3, second-order variations around the thermodynamic equilibrium are challenging to compute. Given that this is not directly related to the main result of the present paper, we refer to Appendix B in [6] for a detailed calculation.

3.3. Thermodynamic Blocking

Let us now investigate the possibility of thermodynamic blocking in 3D. Following the same approach as in 1D (Section 2.4), let us assume that our system evolves (slowly) through a series of hydrostatic equilibria, i.e., through a series of states $(\mathbf{p},\mathbf{m})$ that instantaneously comply with Equation (43). We use the pair $(\mathbf{p},\mathbf{m})$ instead of $(\mathbf{u},\mathbf{m})$ to track the system’s state: this eases the upcoming calculations. Let us now assume that our system is in hydrostatic equilibrium. Given the constraint from Equation (43), we can describe the instantaneous state of the system only using m. Combining the relations from Equations (45) and (46), we can rewrite the hydrostatic equilibrium condition as

$$p^{\prime }\left(r\right)=-{\displaystyle \frac{\mathcal{G}m\left(r\right)m^{\prime }\left(r\right)}{4\pi r^4}}.$$

(59)

Here, we point out that the functional relation between p and m cannot be given in a closed form, contrary to the 1D case, where we could express $\mathbf{u}$ explicitly as a function of $\mathbf{g}$ (see Equation (11)). This makes the 3D calculation slightly less transparent, but it does not affect the results to be derived.

Following Equation (48), the total entropy now becomes a sole function of the enclosed mass profiles, m. It reads

$$S_{\mathrm{A}}[\mathbf{m}]=\int \mathrm{d}\mathbf{r}{m}^{\prime }\left(r\right)\left[\frac{3}{2}\ln \left[p\left(r\right)\right]-\frac{5}{2}\ln \left[\rho \left(r\right)\right]\right],$$

(60)

where we stress that both the pressure and density profiles are to be interpreted as functionals of the enclosed mass profiles, i.e., one has p(r) = p [m](r) (via Equation (59)) and p(r) = p [m](r (via Equation (45)). Similarly, following Equation (61), the total energy now reads

$$E_{\mathrm{A}}[\mathbf{m}]=\int \mathrm{d}\mathbf{r}\left[\frac{3}{2}4\pi r^{2}p\left(r\right)-{\displaystyle \frac{\mathcal{G}m\left(r\right)m^{\prime }\left(r\right)}{r}}\right],$$

(61)

with the same self-consistent constraint to be satisfied by p(r). As previously, the subscript “|_A” emphasises that we are considering systems in hydrostatic equilibrium.

Just like we did in 1D, we now assume that the system is submitted to some (spherically symmetric) adiabatic perturbations, m→m +ϵΔm. Following Equation (34), the variations in the total entropy read

$$\Delta S_{\mathrm{A}}=\int \mathrm{d}r{\displaystyle \frac{\delta S_{\mathrm{A}}}{\delta m\left(r\right)}}\Delta m\left(r\right),$$

(62a)

$${\Delta }^2{S}_{\mathrm{A}}=\int \mathrm{d}r\mathrm{d}\tilde{r}{\displaystyle \frac{{\delta }^2{S}_{\mathrm{A}}}{\delta m\left(t\right)\delta m\left(\tilde{r}\right)}}\Delta m\left(r\right)\Delta m\left(\tilde{r}\right).$$

(62b)

As detailed in Appendix F, we find that the variations in the total entropy are generically given by

$$\Delta S_{\mathrm{A}}=\frac{5}{2}\int \mathrm{d}r{\displaystyle \frac{{\rho }^{\prime }\left(r\right)}{\rho \left(r\right)}}\Delta m\left(r\right),$$

(63a)

$${\Delta }^2{S}_{\mathrm{A}}=-\frac{5}{2}\int \mathrm{d}r{\displaystyle \frac{{[\Delta m^{\prime }\left(r\right)]}^2}{m^{\prime }\left(r\right)}},$$

(63b)

while that of the total energy are given by

$$\Delta E_{\mathrm{A}}=-\frac{5}{2}\mathcal{G}\int \mathrm{d}r{\displaystyle \frac{m\left(r\right)}{r^2}}\Delta m\left(r\right).$$

(64)

As in 1D, we first perform a sanity check of our calculations by considering the case of a thermodynamical equilibrium, undergoing adiabatic perturbations. Imposing the constraints from Equation (53), we find from Equations (63a) and (64) that, at first order, the variations in the total entropy and total energy are connected via

$$\Delta S_{\mathrm{A}}{|}_{\mathrm{T}}={\displaystyle \frac{3}{2u_0}}\Delta E_{\mathrm{A}}{|}_{\mathrm{T}}.$$

(65)

As could have been expected, this relation is identical to the one obtained in Equation (57), where we considered arbitrary perturbations. Again, energy conservation leads to

$$\Delta S_{\mathrm{A}}{|}_{\mathrm{T}}=0.$$

(66)

We now move to the case of systems submitted to adiabatic (and spherically symmetric) perturbations. In that view, let us therefore consider 3D homogeneous self-gravitating systems in hydrostatic equilibrium. Phrased differently, we consider systems characterised by some m, such that

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\displaystyle \rho =\mathrm{cst}}\hfill & (\mathrm{Homogeneous}\mathrm{density}),\hfill \\ {\displaystyle p^{\prime }=\rho g}\hfill & (\mathrm{Hydrostatic}\mathrm{equilibrium}-\mathrm{Equation}\left(43\right)).\hfill \end{array}\right.\end{array}$$

(67)

In practice, homogeneous equilibria of finite extent can be constructed via an “Eddington inversion” (see, e.g., [1]). Such a distribution is a dynamically stable collisionless equilibrium, wherein, in the absence of fluctuations, the velocity dispersion exactly balances the tendency for gravitational collapse.

In a homogeneous system of finite extent, since ${\rho }^{\prime }=0$, we readily find from Equation (63a) that the first-order variation of the entropy vanishes. In addition, recalling from Equation (45) that $m^{\prime }\propto \rho >0$, we find that the second-order variation of the entropy is negative. As a consequence, we therefore have

$$\Delta S_{\mathrm{A}}{|}_{\mathrm{H}}=0;{\Delta }^2{S}_{\mathrm{A}}{|}_{\mathrm{H}}<0.$$

(68)

This is the main result of this section. It states that 3D homogeneous systems in hydrostatic equilibrium are entropy extrema with respect to (spherically symmetric) adiabatic perturbations, i.e., perturbations that maintain the hydrostatic equilibrium. This is another example of thermodynamic blocking. All the discussion elements presented in the 1D case (after Equation (41)) also apply to the 3D case, except for the derivation based on barycenter conservation (Appendix D), since, in 3D, spherically symmetric perturbations do not drive any change in the barycenter. As a final remark, we point out that the conclusion from Equation (68) would still hold if one was to replace the hydrostatic condition from Equation (43) with some arbitrary relation, $p^{\prime }=p^{\prime }\left[\mathbf{m}\right]\left(r\right)$. However, such a constraint would not be physically motivated and is hence of limited relevance. Indeed, kinetic theory teaches us that it is the hydrostatic equilibrium that is to be maintained during relaxation (see the beginning of Section 2.4).

4. Discussion and Conclusions

In this work we used thermodynamics to investigate the dynamics of self-gravitating systems. We successfully reproduced the well-known result that thermodynamic equilibria are entropy extrema, thus validating the formalism. The main results of this paper are Equations (41) and (68), which show that the variation of the thermodynamic entropy of a self-gravitating system is zero if it is homogeneous and submitted to adiabatic perturbations, i.e., perturbations that always comply with the hydrostatic equilibrium. These entropy extrema were also shown to be maxima.

Such distributions are good models of the cores of self-gravitating systems (see, e.g., [22]). The present results show that these homogeneous cores are more stable against internal (noise-driven) perturbations than other hydrostatic configurations. As we have shown, if a self-gravitating system is initially homogeneous and in hydrostatic equilibrium, it will remain so for some long duration. Indeed, at leading order, adiabatic perturbations, i.e., perturbations that maintain the hydrostatic equilibrium, are unable to make it evolve away from this homogeneous state.

The present derivation relies heavily on the existence of a self-consistent relation between the profiles u and g (resp. p and m) in the 1D (resp. 3D) case. Such a functional relation (be it implicit or explicit) between the fields defining the total entropy in Equation (18) (resp. Equation (48)) is indeed essential. Mathematically, they define a “line” in the function space of physical states, the line of hydrostatic equilibria. We have shown that homogeneous systems in hydrostatic equilibrium are entropy maxima along this particular line. In other words, homogeneous distributions and hydrostatic equilibria are necessary, as well as sufficient, conditions for the thermodynamic blocking to occur.

Physically, the present result implies that the redistribution of matter and energy cannot increase the (thermodynamic) entropy of a homogeneous self-gravitating system if hydrostatic equilibrium must be maintained. Phrased differently, in such homogeneous self-gravitating distributions, matter and heat do not flow. We stress however that this does not imply that entropy does not grow at all in these systems. Indeed, placing ourselves within a thermodynamical formalism, we assumed that $S$ was a functional of only two moments of the phase space distribution. As such, the thermodynamic entropy depends only on the matter and kinetic energy densities, namely $\rho =\int \mathrm{d}vf$ and $\rho T=\int \mathrm{d}vf{v}^2$. Of course, the full dynamical entropy depends on infinitely more higher moments, e.g., $\int \mathrm{d}vf{v}^3$, etc. Such moments do not have a direct macroscopic meaning and hence are out of the scope of thermodynamics. Phrased differently, although the present thermodynamic entropy $S[\mathbf{g},\mathbf{u}]$ is frozen in homogeneous systems, the kinetic entropy $S\left[f\right(x,v\left)\right]$ is allowed to grow. We now conclude by listing a few possible avenues for future works.

Extended entropies. In the present work, we limited ourselves to the usual thermodynamic entropy that involves only two moments of the phase space distribution function, namely the matter density and the kinetic energy density. It would be interesting to investigate improved versions of thermodynamics, in which entropy is expressed by more, but still finitely many, moments of the phase space distribution function. In particular, one should determine whether the present thermodynamic blocking also holds within such an extended framework.

Second-order variations. Throughout the present work, when considering thermodynamical equilibria, we repeatedly faced the difficulty of proving that their second-order entropy variation is negative. Taking inspiration from the method leveraged in [6], it would be rewarding to put these proofs on firmer grounds. In particular, one should investigate changes that arise when restricting oneself to adiabatic perturbations.

Beyond spherically symmetric perturbations. In Section 3, when considering 3D systems, we imposed spherical symmetry to the allowed perturbations. Naturally, it would be enlightening to go beyond this assumption. In practice, we expect for the thermodynamical blocking to still hold in that more generic case, though the involved computations should prove more cumbersome.

Numerical exploration in 1D. The dynamics of systems driven by 1D gravity can be exactly integrated (up to round-off errors) using an event-driven scheme [52]. It would therefore be interesting to perform (very) long-term simulations of finite-$N$ 1D homogeneous systems, to quantify how much their self-consistent relaxation toward the thermodynamical equilibrium is delayed. Such simulations would be challenging given that typical (inhomogeneous) quasi-stationary systems already require extensive integration times to reach their thermodynamical equilibrium [29]. To prevent the accumulation of too-large round-off errors, such a numerical exploration would likely require the use of high-precision floating-point computations.

Numerical exploration in 3D. Similarly, one should also numerically investigate the practical impact of thermodynamic blocking in the relaxation of 3D homogeneous spheres. This was already considered in [47], which emphasised that (i) homogeneous spheres undergo more coherent and longer-lasting fluctuations than their inhomogeneous counterparts; (ii) individual particles in homogeneous spheres typically diffuse on a timescale of order $N^{-1/2}{T}_{\mathrm{dyn}}$, with $T_{\mathrm{dyn}}$ the dynamical time, while in inhomogeneous spheres, individual particles diffuse on a timescale of order $N^{-1}{T}_{\mathrm{dyn}}$; but (iii) the overall relaxation of homogeneous spheres, as a whole, is greatly delayed compared to the one of inhomogeneous spheres. Naturally, one ought to revisit, both numerically and analytically, these different trends in the light of the process of thermodynamic blocking.

2D gravity. In the present work, we limited ourselves to presenting the thermodynamic blocking in 1D and 3D. We expect for this process to also occur in 2D self-gravitating systems. Given that 2D gravity shares deep connexions with the dynamics of 2D point vortices (see, e.g., [16,53,54]), it would be worthwhile to explore further the impact of thermodynamic blocking in the hydrodynamic context. In that case, one may rely on efficient symplectic methods to perform robust numerical simulations (see, e.g., [55,56]).

Deviations away from homogeneity. To derive the thermodynamic blocking, we assumed exact homogeneity. In practice, one may wonder how much a given system may differ from homogeneity for the thermodynamic blocking to still hold. Phrased differently, one should determine the width of the domain of near-homogeneous systems whose dynamics remains greatly affected by the thermodynamic blocking. This should prove particularly important for finite-$N$ systems, in which Poisson fluctuations are unavoidable, i.e., in which exact homogeneity is never truly satisfied. Finally, realistic homogeneous cores are usually embedded in a more extended (possibly infinite) halo with outward decreasing density (see, e.g., [50]). One should revisit the present calculations while accounting for such an extended environment, given its possible impact on the vanishing of perturbations at the system’s boundaries.

Appendix A.1. 1D Energy

In this Appendix, we present a heuristic derivation of the total energy of a 1D self-gravitating system. For the kinetic (or thermal) energy, following the definition from Equation (7) we have

$$\begin{array}{cc}\hfill E_{\mathrm{kin}}& =\int \mathrm{d}x\rho (x)u(x)\hfill \\ \hfill & =-{\displaystyle \frac{1}{2\mathcal{G}}}\int \mathrm{d}x{g}^{\prime }\left(x\right)u\left(x\right).\hfill \end{array}$$

(A1)

To compute the 1D gravitational energy, one ought to be careful since the 1D gravitational interaction potential does not decay to zero for infinite separation, contrary to the 3D case. We start from Poisson equation (Equation (8)). Integrating both sides over the domain $[-x,x]$, we find

$$m\left(x\right)=-g\left(x\right)/\mathcal{G},$$

(A2)

with $m\left(x\right)$ the mass enclosed within that domain. Now, let us imagine the build-up of a 1D self-gravitating system by consecutively adding some small mass element, $\mathrm{d}m$ to it, at location $x$. In practice, in order to preserve the symmetry during this process, we add $\mathrm{d}m/2$ to both the positive and negative edges, so as to keep the system’s centre in $x=0$. Doing so, we find that the system’s total gravitational energy is

$$E_{\mathrm{gr}}={\int }_0^{+\infty }{\displaystyle \frac{\mathrm{d}m}{2}}\mathcal{G}\mathrm{mx}-{\int }_0^{-\infty }{\displaystyle \frac{\mathrm{d}m}{2}}\mathcal{G}\mathrm{mx},$$

(A3)

where the minus sign in the second term accounts for the fact that the gravitational acceleration, $g$, points in opposite direction on the positive and negative sides. In addition, for the potential at the edges during this build-up, we used $\varphi \left(x\right)=\mathcal{G}mx$. This follows from a further integration of Equation (8). Using Equation (A2), we can now rewrite Equation (A3) solely as a function of $g$. We find

$$\begin{array}{cc}\hfill E_{\mathrm{gr}}& ={\displaystyle \frac{1}{2\mathcal{G}}}{\int }_{-\infty }^{+\infty }\mathrm{d}xg\left(x\right)g^{\prime }\left(x\right)x\hfill \\ \hfill & ={\displaystyle \frac{1}{2\mathcal{G}}}\left({\left[{\displaystyle \frac{{\left[g\left(x\right)\right]}^2}{2}}x\right]}_{-\infty }^{+\infty }-{\int }_{-\infty }^{+\infty }\mathrm{d}x{\displaystyle \frac{{\left[g\left(x\right)\right]}^2}{2}}\right).\hfill \end{array}$$

(A4)

We note that the boundary term in the last equality is irrelevant when calculating the functional derivatives with respect to $g$. It is necessary however to lend a finite value to $E_{\mathrm{gr}}$. Gathering the kinetic energy, $E_{\mathrm{kin}}$ (Equation (A1)), with the gravitational energy, $E_{\mathrm{gr}}$ (Equation (A4)), we recover the expression of the total energy given in Equation (19).

Appendix A.2. 3D Energy

We now consider the case of a 3D self-gravitating system. For the kinetic energy, we write

$$E_{\mathrm{kin}}=\int \mathrm{d}\mathbf{r}\rho \left(\mathbf{r}\right)u\left(\mathbf{r}\right).$$

(A5)

Using the law of ideal gas from Equation (42), along with the relation from Equation (45), this becomes

$$E_{\mathrm{kin}}=\int \mathrm{d}r\frac{3}{2}4\pi r^{2}p\left(r\right).$$

(A6)

To compute the gravitational energy, we start from

$$E_{\mathrm{gr}}=-\frac{1}{2}\int \mathrm{d}\mathbf{r}\mathrm{d}\tilde{\mathbf{r}}{\displaystyle \frac{G\rho \left(\mathbf{r}\right)\rho \left(\tilde{\mathbf{r}}\right)}{|\mathbf{r}-\tilde{\mathbf{r}}|}},$$

(A7)

where the factor $\frac{1}{2}$ prevents over-counting. Performing the usual Legendre expansion of the gravitational potential [1], and leveraging spherical symmetry, this becomes

$$\begin{array}{cc}\hfill E_{\mathrm{gr}}& =-{\displaystyle \frac{\mathcal{G}}{2}}\int \mathrm{d}r\mathrm{d}\tilde{r}4\pi r^{2}4\pi {\tilde{r}}^2{\displaystyle \frac{\rho \left(r\right)\rho \left(\tilde{r}\right)}{\mathrm{Max}[r,\tilde{r}]}}\hfill \\ \hfill & =-{\displaystyle \frac{\mathcal{G}}{2}}\int \mathrm{d}r\mathrm{d}\tilde{r}{\displaystyle \frac{m^{\prime }\left(r\right)m^{\prime }\left(\tilde{r}\right)}{\mathrm{Max}[r,\tilde{r}]}},\hfill \end{array}$$

(A8)

where we used the relation from Equation (45). Since the integrand from Equation (A8) is symmetric with respect to $r\leftrightarrow \tilde{r}$, it suffices to compute the integral from Equation (A8) in the domain $0\le \tilde{r}\le r<+\infty$, and double it. Equation (A8) becomes

$$\begin{array}{cc}\hfill E_{\mathrm{gr}}& =-\mathcal{G}{\int }_0^{+\infty }\mathrm{d}r{m}^{\prime }\left(r\right){\int }_0^r\mathrm{d}\tilde{r}{\displaystyle \frac{m^{\prime }\left(\tilde{r}\right)}{r}}\hfill \\ \hfill & =-\mathcal{G}{\int }_0^{+\infty }\mathrm{d}r{\displaystyle \frac{m^{\prime }\left(r\right)m\left(r\right)}{r}},\hfill \end{array}$$

(A9)

where we used the fact that $m(r=0)=0$. Gathering the kinetic energy, $E_{\mathrm{kin}}$ (Equation (A6)), with the gravitational energy, $E_{\mathrm{gr}}$ (Equation (A9)), we recover the expression of the total energy functional given in Equation (49).