Gaussian Scrooge Distribution

• Gaussian Scrooge distribution is a universal probability measure on quantum pure states defined via a Gaussian → adjust → project procedure, uniting features of Gaussian and Cauchy laws.
• It is constructed to ensure that the ensemble average reproduces the canonical density matrix, providing rigorous underpinnings for thermal equilibrium in quantum subsystems.
• The framework extends to systems with spin and has applications in laser physics and quantitative finance, effectively capturing both central tendencies and fat-tail behaviors.

The Gaussian Scrooge distribution—also termed the Gaussian Adjusted Projected (GAP) measure—is a universal, mathematically rigorous probability measure on the pure states (wave functions) of quantum subsystems in thermal equilibrium. Its construction unites key features of Gaussian and Cauchy distributions, and its role is foundational in quantum statistical mechanics, especially for characterizing the equilibrium behavior of subsystems weakly coupled to large environments. The GAP approach also illuminates the conditional wave function’s equilibrium distribution in high-dimensional Hilbert spaces and extends naturally to mixed systems including spin degrees of freedom.

1. Mathematical Definition and Constructions

Let $H$ be a finite-dimensional complex Hilbert space of dimension $n$, and let $\rho$ be a density matrix on $H$ ($\rho \geq 0$, $\tr\rho = 1$). The GAP measure, denoted $GAP(\rho)$, is a probability measure on the unit sphere $S(H) = \{\psi\in H: \|\psi\|=1\}$ with the following equivalent constructions:

1. Gaussian → Adjust → Project Workflow:
   • Gaussian Ensemble $G(\rho)$: Diagonalize $\rho = \sum_j p_j |j\rangle\langle j|$. Generate $X_j$ as independent complex Gaussian random variables with $\mathbb{E}[X_j]=0$, $\mathbb{E}[|X_j|^2]=p_j$. The vector $\Psi^{G(\rho)} = \sum_j X_j|j\rangle$ is Gaussian with covariance $\rho$.
   • Adjustment: Reweight $G(\rho)$ by $\|\psi\|^2$ to obtain $GA(\rho)(d\psi) = \|\psi\|^2 G(\rho)(d\psi)$, ensuring normalization.
   • Projection: Project radially onto $S(H)$: $$\Psi^{GAP(\rho)} = \Psi^{GA(\rho)}/\|\Psi^{GA(\rho)}\|$$.
2. Alternative via Uniform Sphere:
   • Draw $\Psi^u$ uniformly from $S(H)$; set $\Psi^{D(\rho)} = \sqrt{n\rho}\Psi^u$. Adjust and project as above. The resulting distribution equals $GAP(\rho)$.
3. Purification Characterization:
   • Construct a purification $\Phi \in S(H\otimes H_2)$ with $\text{tr}_2 |\Phi\rangle\langle\Phi| = \rho$. Sample $\Psi_2\in S(H_2)$ using $$\mu_2(d\psi_2) = n\,||\langle\psi_2|\Phi\rangle||^2 u_2(d\psi_2)$$. Then $$\Psi = \langle\Psi_2|\Phi\rangle / \|\langle\Psi_2|\Phi\rangle\| \in S(H)$$ has distribution $GAP(\rho)$ (Pandya et al., 2013).

The GAP measure preserves the property that the induced mixed state is $\rho$: $$\int_{S(H)} GAP(\rho)(d\psi) |\psi\rangle\langle\psi| = \rho$$.

2. GAP Measure as the Subsystem Equilibrium Law

Given a system $S$ weakly coupled to a large bath $B$, the joint pure state of $S\cup B$ with total energy constrained to an interval $[E, E+\delta]$ typically yields a reduced state on $S$ close to the canonical density matrix:

$\rho_\beta^S = \frac{1}{Z} e^{-\beta H_S}$

(Canonical typicality; $\beta$ fixed by energy per degree of freedom). The conditional wave function of $S$—constructed by partial inner product over a basis in $H_B$—has a distribution $\mu_S^{\text{cond}}$ that, for "most" joint $\Psi$ and basis choices, is:

$\mu_S^{\text{cond}} \approx GAP(\rho_\beta^S)$

The emergence of the GAP distribution here is a consequence of high-dimensional concentration of measure and the hereditary property of GAP under partial tracing (Pandya et al., 2013).

3. Relation to Canonical Density Matrix and Marginals

A central property of the GAP measure is that its statistical marginal reproduces the canonical ensemble:

$$\mathbb{E}_{\psi \sim GAP(\rho)} [|\psi\rangle\langle\psi|] = \rho$$

This ensures thermal equilibrium consistency: in the GAP ensemble, the average state is the canonical density matrix, yet the full measure encodes fluctuations at the pure-state level. This result provides rigor to the notion that subsystems possess "random" wave functions in thermal equilibrium, with the GAP law as the appropriate invariant measure (Pandya et al., 2013).

4. Extension to Systems with Spin: Conditional Density Matrix

For systems featuring spin, the naive position-basis-based conditional wave function is inadequate due to nontrivial bath Hilbert space structure. Decompose the system as:

$H_S = H_x \otimes H_r,\quad H_B = H_y \otimes H_s$

where $H_x, H_y$ are spatial components and $H_r, H_s$ encode spin. The global pure state is $\Psi \in H_x\otimes H_r \otimes H_y\otimes H_s$.

• Conditional Wave Function of $S\cup s$: Select $|Y\rangle \in H_y$ randomly (weighted by $||\langle Y|\Psi\rangle||^2$), set $$\psi_{S\cup s}^{\text{cond}} = \langle Y|\Psi\rangle / \|\langle Y|\Psi\rangle\|$$.
• Conditional Density Matrix of $S$: Trace out the bath spin $s$:

$$\rho_S^{\text{cond}} = \text{tr}_s \left(|\psi_{S\cup s}^{\text{cond}}\rangle\langle\psi_{S\cup s}^{\text{cond}}|\right)$$

For most $\Psi$ and bases in $H_y$, the distribution of $\rho_S^{\text{cond}}$ is sharply peaked at $\rho_\beta^S$, with variance exponentially small in the bath size. Thus, in the presence of spin, the conditional density matrix is (essentially) deterministic and equal to the canonical state (Pandya et al., 2013).

5. Distributional Properties and Interpolation between Gaussian and Cauchy Laws

An explicit version of the Gaussian–Scrooge (intermediate) law, introduced in (Liu et al., 2012), provides an explicit PDF that continuously interpolates between Gaussian and Cauchy distributions. For real variables $X$: $$p(x;\mu,\sigma,\nu) = \frac{1}{\nu\pi \exp(\sigma^2-\sigma^2/\nu^2)} \int_0^1 \cos\left(\frac{x-\mu}{\nu} \ln t \right)t^{\sigma^2/\nu^2-1} \exp[(\sigma^2-\sigma^2/\nu^2)t]\,dt$$ Parameters: $\mu$ (location), $\sigma$ (scale), $\nu$ (interpolation).

• As $\nu\to 0^+$, the law converges to Gaussian: $$p(x;\mu,\sigma,0)=\frac1{\sqrt{2\pi}\,\sigma}\exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right)$$
• As $\nu\to 1$, the law becomes Cauchy: $$p(x;\mu,\sigma,1)=\frac{\sigma^2/\pi}{\sigma^4+(x-\mu)^2}$$ The tails are Cauchy-like ($\sim 1/x^2$), precluding the existence of ordinary moments for $\nu>0$ and $m\geq 1$ (Liu et al., 2012).

6. Weighted Moments and Tail Behavior

To address diverging moments caused by fat power-law tails, weighted moments are introduced: $$M_{2n}^{(w)} = \int_{-\infty}^{+\infty} x^{2n} w(x) p(x;0,\sigma,\nu)\,dx$$ Two classes of weight functions yield analytic, convergent representations:

• Cut-off weight: $w_{\text{cut}}(x;R) = \mathbf{1}_{|x| \leq R}$
• Exponential weight: $w_{\text{exp}}(x;\gamma) = e^{-\gamma x^2}$

These yield finite moments for all $\nu$, recover standard Gaussian moments as $\nu\to 0$, and possess closed-form series in hypergeometric functions. When these weights are removed, divergence reemerges, reflecting the underlying $1/x^2$ tail behavior (Liu et al., 2012).

7. Applications in Laser Physics and Quantitative Finance

Spectral-Line Broadening

In laser physics, spectral line shapes are influenced both by homogeneous broadening (leading to Lorentzian/Cauchy profiles) and inhomogeneous Doppler broadening (yielding Gaussian profiles). The intermediate law $p(x;\mu, \sigma, \nu)$ provides a natural one-parameter family interpolating between these physical broadening mechanisms and is fit to experimental lines by adjusting $(\sigma,\nu)$ (Liu et al., 2012).

Stock Return Modeling

Empirical distributions of financial log-returns display leptokurtic peaks at zero with slower-decaying algebraic tails compared to a Gaussian. The intermediate law enables a fit to both the center and tails of return histograms, often finding $0 < \nu < 1$ and yielding superior representations compared to individual Gaussian or Cauchy models, and that are competitive with $q$-Gaussian approaches (Liu et al., 2012).

---

References:

• Tong Liu, Ping Zhang, W.-S. Dai, Mi Xie, "An intermediate distribution between Gaussian and Cauchy distributions" (Liu et al., 2012).
• S. Goldstein, R. Tumulka, N. Zanghì, "Spin and the Thermal Equilibrium Distribution of Wave Functions" (Pandya et al., 2013).