Invariant Boltzmann-Gibbs Distribution

• Invariant Boltzmann–Gibbs distribution is defined as the unique equilibrium law emerging from invariance, independence, and conservation principles.
• Mathematical proofs employ convolution semigroups and support-preserving endomorphisms, establishing exponential tilting as the sole mechanism for invariance.
• This distribution underpins diverse models from statistical mechanics to economic exchange, demonstrating robustness in both discrete and continuous systems.

The invariant Boltzmann–Gibbs distribution is a central construct in statistical mechanics, ensemble theory, and stochastic exchange models, emerging as the unique equilibrium law under general invariance, independence, and thermodynamic consistency requirements. It characterizes the statistical distribution of microstates—whether in physical systems, idealized wealth models, or abstract algebraic settings—where probability weights are exponentially tilted by relevant conserved quantities (e.g., energy, particle number), and its uniqueness stems from the preservation of system composition rules and support. The distribution embodies deep mathematical links across statistical mechanics, probability theory, and algebraic structures.

1. Formal Foundations: Convolution Semigroups and Support-Preserving Endomorphisms

Let $P(\mathbb{N})$ denote the space of finitely supported probability measures on the non-negative integers. The law of the sum of two independent $\mathbb{N}$-valued random variables with respective laws $\mu$, $\nu$ is given by their convolution:

$$(\mu \ast \nu)(m) = \sum_{i + j = m} \mu(i)\nu(j),\qquad m \in \mathbb{N}.$$

This operation endows $P(\mathbb{N})$ with the structure of a commutative semigroup with identity $\delta_0$.

An endomorphism $F: P(\mathbb{N}) \to P(\mathbb{N})$ is support-preserving if $\text{supp}(F(\mu)) = \text{supp}(\mu)$, i.e., $F(\mu)(n)>0$ if and only if $\mu(n)>0$. Sandomirskiy and Tamuz proved that any such $F$ commuting with convolution is necessarily an exponential tilting:

$$(T_\varphi\mu)(n) = \frac{e^{-\varphi(n)} \mu(n)}{\sum_{k} e^{-\varphi(k)} \mu(k)},$$

for some function $\varphi: \mathbb{N} \to \mathbb{R}$; that is,

$F(\mu)(n) \propto e^{-\varphi(n)} \mu(n).$

This characterization establishes exponential tiltings as the only invariant laws under independent composition of subsystems, providing an abstract algebraic foundation for the Boltzmann–Gibbs distribution (Sandomirskiy et al., 2023).

2. Thermodynamic Consistency and Ensemble Theory

Consider a system of microstates $\Omega$ parameterized by external variables and ensemble variables (temperature $T$, generalized forces $X_1,\ldots,X_n$). The probability density over microstates is taken of the generic form

$$\mathrm{Pr}(\omega) \propto f(E(\omega), x_1(\omega), \ldots, x_n(\omega); T, X_1, \ldots, X_n),$$

without preassigning $f$ or maximizing entropy.

Thermodynamic consistency is imposed as follows:

• First law/Legendre structure: There exists an entropy $S$ such that

$$dU = T\,dS + \sum_{\eta=1}^n X_\eta\,d\chi_\eta + \text{other terms}$$

• Ensemble averages directly yield state functions.
• Infinite temperature uniformity: As $T \to \infty$, all microstates become equiprobable.

Gao demonstrated that under these constraints, $f$ must be exponential in a linear combination of conserved quantities:

$$\mathrm{Pr}(\omega) = \frac{1}{Z} \exp\left[\sum_{\eta=1}^n \frac{X_\eta x_\eta^{(\omega)}}{k_BT} - \frac{E^{(\omega)}}{k_BT}\right],\qquad Z = \sum_{\omega} \exp\left[\cdots\right].$$

This law covers canonical, grand-canonical, and isothermal–isobaric ensembles, and is uniquely invariant under the Maxwell relations and Legendre transforms required by thermodynamics (Gao, 2020).

3. Markov Exchange Models and Discrete Analogs

In kinetic theory and stochastic exchange models (such as unbiased dollar transfer among agents), the analog of energy is wealth, and the equilibrium wealth distribution approaches a geometric or exponential law. For finite agents with total conserved wealth the generator is

$$\mathcal{L}_N f(S) = \frac{1}{N}\sum_{i \neq j} \mathbf{1}_{S_i \ge 1}\big[f(\ldots, S_i-1, S_j+1, \ldots) - f(S)\big].$$

In the mean-field limit and large $N$, the agent wealth distribution $p_n(t)$ satisfies nonlinear ODEs and converges to the unique geometric equilibrium

$$p_n^* = (1 - r^*) (r^*)^n,\qquad r^* = \frac{\mu}{1+\mu},\qquad p_0^* = \frac{1}{1+\mu}.$$

The geometric law is the discrete Boltzmann–Gibbs invariant in this context. Quantitative entropy methods yield almost-exponential convergence rates in relative entropy to equilibrium (Cao et al., 2022).

4. Rigorous Graph-Based Exchange Proofs

For economic agents on the vertices of finite connected graphs, one-dollar exchanges yield a time-reversible, irreducible Markov chain with unique stationary law uniformly distributed on microstate space:

$$S_{N,M} = \left\{\xi: V \to \{0,1,2,\ldots\}: \sum_{x \in V} \xi(x) = M\right\}.$$

Marginal stationary distributions converge to

$$\pi\{\xi(x) = d\} = \frac{\binom{M+N-d-2}{N-2}}{\binom{M+N-1}{N-1}},$$

and in the thermodynamic limit:

$$\lim_{N \to \infty} \pi\{\xi(x) = d\} = \frac{1}{T} e^{-d/T},$$

where $T = M/N$ is the "money temperature". This result confirms the exponential (Boltzmann–Gibbs) law as the unique invariant equilibrium distribution regardless of graph geometry (Lanchier, 2017).

5. Uniqueness and Universality

The invariance of the Boltzmann–Gibbs distribution under independent composition and support preservation underlies its universality. In algebraic terms, any law on $P(\mathbb{N})$ that (i) preserves which energies can occur, and (ii) respects series composition of uncoupled systems (convolution of measures), is necessarily an exponential tilting of the form described above (Sandomirskiy et al., 2023).

In thermodynamic ensembles, the invariant Boltzmann–Gibbs law is the sole probability measure consistent with all Maxwell relations, Legendre structure, and infinite-temperature uniformity (Gao, 2020). In stochastic exchange and financial models, this distribution emerges as the unique stationary solution under unbiased local conservation (Cao et al., 2022, Lanchier, 2017).

6. Discrete vs. Continuous Realizations and Alternative Models

The discrete exponential law encountered in money-exchange and graph models corresponds to the continuous Boltzmann–Gibbs law for energy distributions:

• Discrete (geometric/exponential in integer state): $p_n^* \propto (r^*)^n$
• Continuous (canonical exponential in energy): $p(E) \propto e^{-\beta E}$

For size-biased interaction on graphs, limiting distributions become Poissonian. The exponential invariant remains robust against variations in topology, interaction scope, or system size, confirming its deep connection to statistical equilibrium and invariance principles (Lanchier, 2017).

7. Concluding Observations

The invariant Boltzmann–Gibbs distribution is singled out by the interplay of independence, invariance under subsystem composition, and support preservation—not by appeal to entropy maximization or prior distributions. Its emergence across physical, algebraic, and economic models signifies a deep unification of probabilistic, thermodynamic, and combinatorial structures. All rigorous derivations cited above confirm this inexorable necessity in describing equilibrium states for broad classes of systems.