Relative tightness of two classical thermodynamic concentration inequalities

Determine, for time-independent classical Markov processes, which of the two lower bounds on the probability P(N(τ)=0) is tighter in general: (i) the bound cos[(1/2) ∫_0^τ (√A(t)/t) dt]^2 ≤ P(N(τ)=0) derived from the classical dynamical activity A(t), and (ii) the bound exp(−𝔞(0) τ) ≤ P(N(τ)=0) expressed in terms of the instantaneous dynamical activity rate 𝔞(0).

Background

The paper derives two concentration-type lower bounds on the probability that a trajectory observable N(τ) vanishes in time-independent classical Markov processes. One bound involves the time-integrated classical dynamical activity A(t) via a cosine function, while the other uses the instantaneous dynamical activity rate at the initial time, 𝔞(0), yielding an exponential form.

While the authors note a practical advantage of the exponential bound holding for any τ>0 and that it is tighter in steady state, they explicitly state that it is unknown, in general, which of the two bounds provides the tighter estimate for P(N(τ)=0). Establishing the general dominance or delineating regimes of tightness remains unresolved.