Thermodynamic Geometric Constraint on the Spectrum of Markov Rate Matrices

TL;DR

The paper proves a universal geometric constraint on the spectrum of Markov rate matrices by formulating the problem in terms of the numerical range W(tilde{R}) of a similarity-transformed generator tilde{R} and decomposing it into symmetric and antisymmetric parts, linking spectrum to correlation function derivatives. The ellipse bound EL, specified by EL = {((Re z + alpha)/alpha)^2 + (Im z /(alpha beta))^2 <= 1} with alpha = max_i |R_{ii}| and beta = tanh(max_e F_e / 2), confines all eigenvalues and yields a bound on oscillation frequencies max_n Im lambda_n <= alpha beta; the bound is tight in cases such as uniform cycles. The work relates this geometric constraint to previous sector bounds and to Uhl-Seifert’s conjecture EL_c, showing EL_c bounds eigenvalues but not the full numerical range, and discusses implications for autocorrelation timing and potential quantum extensions. Overall, the ellipse theorem provides a thermodynamic geometric constraint that ties irreversibility to spectral shape and the initial-time behavior of correlations, offering sharper insight than purely algebraic bounds like Gershgorin.

Abstract

The spectrum of Markov generators encodes physical information beyond simple decay and oscillation, which reflects irreversibility and governs the structure of correlation functions. In this work, we prove an ellipse theorem that provides a universal thermodynamic geometric constraint on the spectrum of Markov rate matrices. The theorem states that all eigenvalues lie within a specific ellipse in the complex plane. In particular, the imaginary parts of the spectrum, which indicate oscillatory modes, are bounded by the maximum thermodynamic force associated with individual transitions. This spectral bound further constrains the initial short-time behavior of correlation functions between two arbitrary observables. Finally, we compare our result with a previously proposed conjecture, which remains an open problem and warrants further investigation.

Figures (5)

• Figure 1: Bounds on the spectrum of Markov rate matrices. The blue line shows the elliptical bound derived in this work, and the open circles denote the eigenvalues. Our theorem states that the spectrum is contained within this ellipse. The major axis lies along the real axis and spans the interval from $-2 \max_i |R_{ii}|$ to $0$, where $|R_{ii}|$ is the escape rate. The minor axis has length $2 \beta \max_i |R_{ii}|$ with $\beta= \tanh \left(\max_e F_e /2\right)$, where $F_e$ denotes the thermodynamic force on edge $e$. The black dashed lines correspond to the sectorial bound reported by Ohga et al. in Ref. Ohga_corsscorr_2023. The opening angle of the sector is bounded above by $2 \arctan{C}$, where $C = \max_c \{ \tanh[\mathcal{F}_c/(2 n_c)]/ \tan(\pi/n_c) \} \leq \max_c \mathcal{F}_c/(2 \pi)$ with $\mathcal{F}_c$ and $n_c$ denoting the thermodynamic force and the length of cycle $c$, respectively. Therefore, the spectrum is restricted to the blue shaded region. The red circle represents the Gershgorin circle bound, and the green dashed ellipse represents the bound numerically conjectured by Uhl and Seifert Uhl_conjecture_2019. Here, the vertical extents of the Gershgorin circle and the ellipse conjectured by Uhl and Seifert are $2 \max_i |R_{ii}|$ and $2 \beta' \max_i |R_{ii}|$, respectively, where $\beta' = \tanh \left( \max_c \{ \mathcal{F}_c /(2n_c) \}\right)$.
• Figure 2: Thermodynamic bound on maximal frequency for cycles with $N=4$ states. The open circles represent the maximum imaginary part of the eigenvalues of rate matrices corresponding to random walks on a ring with random jump rates. The blue line represents the bound given by Eq. \ref{['frequency']}.
• Figure 3: Comparison between the conjectured ellipse $EL_c$ (red line) and the derived bound $EL$ (blue line). This example illustrates that the conjectured ellipse $EL_c$ contains all eigenvalues of $R$ (hollow circles), but does not enclose the numerical range $W(\tilde{R})$ (black shaded region). The bound by Ohga et al. also holds for the numerical range $W(\tilde{R})$. The numerical range is calculated using the algorithm in Ref. Johnson_calculation_1978. This example is obtained from a random walk on a ring (as in Fig. 2) with i.i.d. jump rates sampled uniformly from the interval $(0,1)$. We sample such random walks for increasing ring size $N$. For $N \geq 6$, cases where $EL_c$ does not enclose $W(\tilde{R})$ are observed, and the frequency of such cases increases with $N$. This figure shows a representative instance with $N = 7$, found within 1000 random samples.
• Figure 4: Reduced numerical range and spectral gap. An illustration of the symmetrized Liouvillian gap $g_s$ (green dashed line) given by the maximum real part of the complex number in the reduced numerical range $W_r(\tilde{R})$ (red shaded region). The blue shaded region represents the thermodynamic geometric bound, and the black shaded region shows the numerical range $W(\tilde{R})$. The numerical ranges are calculated using the algorithm in Ref. Johnson_calculation_1978.
• Figure S1: Numerical results for the eigenvalue spectra of rate matrices with various dimensions $n$. We consider only single-cycle cases with random transition rates and illustrate the relative shape of $\Omega_n'$ with respect to Uhl and Seifert's ellipse by mapping $\Omega_n'$ to $\Omega_n$ using the inverse of the transformation in Eq. \ref{['Karpelevich_rate']}. The unit circle is shown in red, the boundary of $\Omega_n$ is shown in blue, and the transformed spectra are shown by hollow circles.