Loschmidt's paradox asks why macroscopic irreversibility is universal despite the time-reversal symmetry of microscopic dynamics. We argue that irreversibility is not a property of the dynamics but of accessibility: chaotic evolution drives phase-space structure below the quantum resolution scale $\ell_\hbar$, at a critical time $t_c = λ^{-1}\ln(δ_0/\ell_\hbar)$, after which the time-reversed microstate exists as a valid solution of Hamilton's equations but cannot be selected by any physically admissible operation. The mechanism operates entirely within the semiclassical regime $t_c \leq t_E$, where classical geometry is exact. This provides a dynamical resolution of the Loschmidt paradox.
The quantum foundation is established using a Krylov-complexity framework: we prove that for any $H(t)=H(-t)$, the quantum Lyapunov exponent satisfies $λ_L^{\rm forward} = λ_L^{\rm backward}$. The arrow of time is not in the dynamics. The mechanism predicts sigmoid fidelity decay, logarithmic scaling of $t_c$ with $λ^{-1}$, and ensemble-size independence of the inaccessibility threshold -- all consistent with three decades of Loschmidt echo experiments and confirmed in a stadium-billiard simulation reported here. Underlying everything: quantum mechanics conserves information exactly. Entropy, defined as the logarithm of the multiplicity $Ω$ -- the number of possibilities consistent with the available information -- can only increase when information becomes operationally inaccessible. The second law reflects not a breakdown of microscopic reversibility, but the dynamical inaccessibility of the information required to reverse it.
Loschmidt's paradox asks why macroscopic irreversibility is universal despite the time-reversal symmetry of microscopic dynamics. We argue that irreversibility is not a property of the dynamics but of accessibility: chaotic evolution drives phase-space structure below the quantum resolution scale ℓℏ, at a critical time tc=λ−1ln(δ0/ℓℏ), after which the time-reversed microstate exists as a valid solution of Hamilton's equations but cannot be selected by any physically admissible operation. The mechanism operates entirely within the semiclassical regime tc≤tE, where classical geometry is exact. This provides a dynamical resolution of the Loschmidt paradox.
The quantum foundation is established using a Krylov-complexity framework: we prove that for any H(t)=H(−t), the quantum Lyapunov exponent satisfies λLforward=λLbackward. The arrow of time is not in the dynamics. The mechanism predicts sigmoid fidelity decay, logarithmic scaling of tc with λ−1, and ensemble-size independence of the inaccessibility threshold -- all consistent with three decades of Loschmidt echo experiments and confirmed in a stadium-billiard simulation reported here. Underlying everything: quantum mechanics conserves information exactly. Entropy, defined as the logarithm of the multiplicity Ω -- the number of possibilities consistent with the available information -- can only increase when information becomes operationally inaccessible. The second law reflects not a breakdown of microscopic reversibility, but the dynamical inaccessibility of the information required to reverse it.
The Loschmidt echo--also known as fidelity--is a very useful tool to study irreversibility in quantum mechanics due to perturbations or imperfections. Many different regimes, as a function of time and strength of the perturbation, have been identified. For chaotic systems, there is a range of perturbation strengths where the decay of the Loschmidt echo is perturbation independent, and given by the classical Lyapunov exponent. But observation of the Lyapunov decay depends strongly on the type of initial state upon which an average is carried out. This dependence can be removed by averaging the fidelity over the Haar measure, and the Lyapunov regime is recovered, as has been shown for quantum maps. In this work, we introduce an analogous quantity for systems with infinite dimensional Hilbert space, in particular the quantum stadium billiard, and we show clearly the universality of the Lyapunov regime.
I give an explanation of the thermodynamic arrow-of-time (namely entropy increases with time) within a quantum mechanical framework. This entails giving a solution to the Loschmidt paradox, i.e. showing how an irreversible macro-dynamics can arise from a reversible micro-dynamics. I argue that, in accordance to the reversible dynamics, both entropy-increasing and entropy-decreasing transformations take place, but entropy-decreasing transformations cannot leave any information of their having happened. This is indistinguishable from their not having happened at all. The second law of thermodynamics is then reduced to a tautology: the only transformations that can be seen are those where entropy does not decrease. However, typicality arguments seem to prevent this argument to be used as a complete solution to the arrow-of-time dilemma: it might still be necessary to postulate a low entropy initial state for the system under consideration.
This work proposes a structural mechanism for the emergence of thearrow of time, based on the Principle of Quantum Desynchronization.We argue that irreversibility does not require the thermodynamiclimit of an infinite number of particles, but rather emerges infinite systems due to spectral incommensurability. For incommensurablesystems—the generic case in nature—exact recurrence is mathematicallyimpossible even with only two components. We further demonstrate thatany minimal error in time inversion makes exact reversal impossible,with the probability of success decaying exponentially with thesquare of the number of components. The framework resolvesLoschmidt's paradox on microscopic, structural, and phenomenologicallevels, giving rise to an updated formulation: Loschmidt's Law,which states that the reversibility of individual components does notimply reversibility of the complete system. The periodic behavior oftime crystals is explained as exceptionally commensurable systems,while gravity—through relativistic time dilation—is identified as thefundamental and unavoidable mechanism guaranteeing spectralincommensurability in real physical systems. The theory offers amechanistic, not merely statistical, solution to the problem oftemporal irreversibility.
We had previously shown [T. Yarman et al., Results Phys. 10 (2018) 818–821] that the classical adiabatic relation pV 5/3=Constant for an ideal monatomic gas admits a direct quantum-mechanical formulation in terms of Planck’s constant and confinement-dependent quantum-state indices. This result suggests that macroscopic thermodynamic scaling laws reflect an underlying quantum-state structure. In the present work, we revisit this quantum-state scaling framework and discuss how it may provide a complementary language for thermodynamic accessibility and macroscopic irreversibility. In particular, we emphasize the conceptual distinction between microscopic reversibility, the arrow of time, and operational macroscopic irreversibility. Within this perspective, irreversibility may be interpreted in terms of asymmetries in accessible quantum-state configurations under changing confinement and energy conditions, while the underlying microscopic equations remain formally time-reversal symmetric. At the same time, the concept of operationally inaccessible microscopic quantum reconstruction, such as that with regards to fission, naturally emerges. Still, we do not claim to replace Boltzmannian, correlation-based, or information-theoretic approaches. Rather, the present study proposes a complementary quantum-state scaling perspective for discussing thermodynamic accessibility and the macroscopic evolution of complex systems. Even so, within the present framework, these practical limitations on thermodynamic accessibility do not necessitate a breakdown of formal microscopic reversibility; rather, they align with the Boltzmannian view that macroscopic reversals remain formally possible yet are rendered overwhelmingly inaccessible in practice.
We present a unified operational framework for the arrow of time grounded in irreversible information loss. While microscopic physical laws are time-reversal invariant, macroscopic irreversibility arises when systems overwrite metastable records. We show that recent reversible quantum-control experiments do not reverse time, but instead operate in regimes where no logically irreversible record is created. To formalize this, we introduce the **forgetting rate** (Ξcr), a coarse-grained rate defined on physically selected metastable record states (e.g., pointer states, memory registers, or macroscopic configurations). Unlike standard entropy production, which is expressed in terms of microscopic transition asymmetry, (Ξcr) captures the **rate of irreversible record overwriting**. In this formulation, entropy production obeys the lower boundS˙irr≥kBΞcr,consistent with Landauer's principle but expressed at the level of record dynamics. Within this framework, thermodynamic, quantum-measurement, psychological, and cosmological arrows emerge as substrate-dependent manifestations of Ξcr>0. Temporal asymmetry is therefore not a single global property, but a distributed feature of physical information-processing systems. The approach naturally explains the decoupling of arrows in composite systems and identifies experimentally accessible regimes where irreversibility can be locally suppressed. This work does not introduce a new microscopic law but provides a unifying, operational reformulation that connects information thermodynamics, decoherence theory, and memory dynamics through a single coarse-grained rate.
The Loschmidt echo (LE) is a measure of the sensitivity of quantum mechanics to perturbations in the evolution operator. It is defined as the overlap of two wave functions evolved from the same initial state but with slightly different Hamiltonians. Thus, it also serves as a quantification of irreversibility in quantum mechanics. In this thesis the LE is studied in systems that have a classical counterpart with dynamical instability, that is, classically chaotic. An analytical treatment that makes use of the semiclassical approximation is presented. It is shown that, under certain regime of the parameters, the LE decays exponentially. Furthermore, for strong enough perturbations, the decay rate is given by the Lyapunov exponent of the classical system. Some particularly interesting examples are given. The analytical results are supported by thorough numerical studies. In addition, some regimes not accessible to the theory are explored, showing that the LE and its Lyapunov regime present the same form of universality ascribed to classical chaos. In a sense, this is evidence that the LE is a robust temporal signature of chaos in the quantum realm. Finally, the relation between the LE and the quantum to classical transition is explored, in particular with the theory of decoherence. Using two different approaches, a semiclassical approximation to Wigner functions and a master equation for the LE, it is shown that the decoherence rate and the decay rate of the LE are equal. The relationship between these quantities results mutually beneficial, in terms of the broader resources of decoherence theory and of the possible experimental realization of the LE.