The efficient manipulation of thermodynamic states within the finite time is fundamentally constrained by the intrinsic dissipative cost. While the slow-driving regime is well-characterized by a universal 1/τ-scaling of irreversibility, the physics governing fast, non-adiabatic transitions remains elusive. Here, we propose the polytropic steering protocols that provide an exact analytical bridge between the isothermal and adiabatic limits for Brownian particles far-from-equilibrium. We demonstrate that for any protocol duration τ, the system can be precisely steered along a prescribed polytropic trajectory, revealing a striking non-monotonic dependence of irreversibility on the driving rate. Contrary to the near-equilibrium paradigm where faster driving necessitates higher energetic costs, we identify a most-irreversible timescale, beyond which dissipation is anomalously suppressed by rapid driving. By mapping these protocols onto a broad class of controllable thermodynamic cycle, we establish power-efficiency tradeoffs and position the polytropic index as a genuine thermodynamic control knob for the rational design of high-speed, high-performance microscopic thermal machines.

Geometric phases are foundational to isolated quantum systems, yet their thermodynamic role in open systems remains unrevealed Developing a dissipative adiabatic perturbation expansion, we discover a Berry-phase-induced chiral work difference that survives decoherence. This chirality evolves from an interferometric thermodynamic Aharonov-Bohm effect in the unitary regime to a fringe-free signal in the dissipative regime. We illustrate this framework in a two-level system and assess its experimental feasibility. Our findings clarify the role of quantum geometry in the geometric formulation of thermodynamics.

Hysteresis, with rich dynamical behaviors-especially in interacting systems-has drawn broad research interest. Yet its dynamic scalings across timescales lack a unified description, and their transitions remain unclear. Here, we study the stochastic ϕ^{4} model driven periodically by an external field H. For large systems with small noise strength σ, we find the coercivity H_{c}≡H(⟨ϕ⟩=0) sequentially exhibits distinct behaviors with increasing driving rate v_{H}: v_{H}-scaling increase, stable plateau (v_{H}^{0}), v_{H}^{1/2}-scaling increase, and abrupt decline to disappearance. The plateau reflects the competition between thermodynamic and quasistatic limits, namely, lim_{σ→0}lim_{v_{H}→0}H_{c}=0, and lim_{v_{H}→0}lim_{σ→0}H_{c}=H^{*}. Here, H^{*} is exactly the field-driven first-order phase transition point. In the post-plateau regime, (H_{c}-H_{P}) scales with (v_{H}-v_{P})^{2/3} with v_{P} and H_{P} being the reference points of the plateau. Moreover, we reveal a finite-size scaling for the coercivity plateau as v_{P}∼σ^{2} and (H^{*}-H_{P})∼σ^{4/3} by utilizing renormalization-group theory. Our Letter provides a panoramic view of finite-time scalings of the hysteresis and offers new insights into the interplay between finite-time and finite-size effects in nonequilibrium systems.

While the impossibility of perfectly identifying non-orthogonal states is a cornerstone of quantum information science, their probabilistic discrimination is nonetheless permissible. Here, we propose a two-reservoir quantum machine driven by this mechanism to map its functional boundaries across the parameter space of the state overlap μ and the Carnot efficiency η_C. Within this η_C-μ plane, the machine exhibits phase-transition-like functional switching among a pure heat-engine phase, a mixed phase, and a dissipative phase. We identify critical thresholds governing these transitions: strong thermal driving (η_C ≥ 0.5) unconditionally guarantees positive work extraction, whereas weak driving (η_C ≲ 0.13) induces an anomalous reentrant transition, where increasing μ unexpectedly restores engine functionality after a purely dissipative regime. Our results explicitly demonstrate how quantum mechanics and thermodynamics jointly constrain information-to-energy conversion.

Battery discharging is governed by a fundamental trade-off between output power and energy conversion efficiency due to internal dissipation. In this paper, we demonstrate that such a trade-off universally yields a parabolic envelope P∝η(1−η). The efficiency at maximum power is exactly one half, mirroring the well-known half-Carnot limit in finite-time thermodynamics. To extend this bound into practical operational rules, we formulate a multistage constant-discharging (MSCD) schedule subject to simultaneous real-time load demands and a global discharging deadline. Analytical resolution via the Karush–Kuhn–Tucker conditions reveals a remarkably compact optimal policy: Ii★=max(Iireq,I0). Under this rule, stages limited by external demand run exactly at their minimum required currents, while all remaining stages are elevated to a uniform baseline I0 fixed by the deadline constraint. By tracing the dissipation–time Pareto front, we quantify how internal resistance shifts the operational boundaries and sharpens the trade-off corner. This analysis establishes a rigorous thermodynamic baseline for the scheduling layer of battery management systems, offering natural extensions to nonlinear models incorporating temperature and state-of-charge dependencies.

Perfect deterministic distinguishing of nonorthogonal quantum states is forbidden by the linear and unitary structure of quantum mechanics. It has often been assumed that, if such distinguishing were available, it would be the resource enabling work extraction from a single heat bath. We show that this expectation identifies the wrong thermodynamic operation and prove such hypothetical operation increases, rather than decreases, the joint entropy of system and detector. The entropy-decreasing resource is instead the inverse operation, which we call nonorthogonal-state erasure. Reanalyzing a Peres-type Szilard engine, we show that the apparent extracted work W_ext=0.2766k_BT for an equal mixture of an atomic ensemble with spin state |↑⟩ and |→⟩. Thus the apparent second-law violation is supplied not by nonorthogonal-state distinguishing, but by a nonorthogonal quantum state erasure.

While externally driven information engines are well understood, the thermodynamic constraints of their autonomous counterparts remain an open question. Here, we investigate the finite-time operation of an autonomous machine functioning as both an information eraser and a refrigerator, revealing that its irreversibility is bounded by the transient information geometry. Beyond steady-state boundaries, we map the landscape of optimal operation times across both functional modes, uncovering a unique synergistic regime where erasure power P and efficiency η increase simultaneously. Fundamentally, this performance is governed by a trade-off relation, v(1-η)P/η≤ D, where v is the operational speed and D denotes an information-geometric distance. Our findings pave the way for optimizing fast autonomous information-energy conversion.

The coercivity landscape for characterizing hysteresis in interacting systems across multiple timescales is proposed by Chen et al. in a companion paper [Phys. Rev. Lett. 136, 117102 (2026)10.1103/5rg8-52gl]. For the stochastic ϕ^{4} model under periodic driving of rate v_{H}, the coercivity landscape H_{c}(v_{H}) exhibits plateau features at a characteristic rate v_{P} with the corresponding coercivity H_{P}. Below this plateau (v_{H}v_{P}), scaling in the fast-driving regime, H_{c}∼v_{H}^{1/2}, is completely different from that, H_{c}-H_{P}∼(v_{H}-v_{P})^{2/3}, in the postplateau slow-driving regime. The emergence of the plateau with a finite-size scaling reflects the competition between the thermodynamic limit and the quasistatic limit. In this paper, we provide detailed analytical proofs and numerical evidence supporting these results. Moreover, to demonstrate the coercivity landscape in concrete physical systems, we study the magnetic hysteresis in the Curie-Weiss model and analyze its finite-size effects. We reveal that finite-time coercivity scaling shows model-specific behavior only in the fast-driving regime, while exhibiting universal characteristics elsewhere.

Thermal metamaterials represent a transformative paradigm in modern physics, synergizing thermodynamic principles with metamaterial engineering to master heat flow at will. As next-generation technologies demand multi-scale thermal control, this field urgently requires systematic frameworks to unify its multidisciplinary advances. Curated through a global collaboration involving over 50 specialists across 25 subdisciplines, this review primarily summarizes two decades of advancements, ranging from theoretical breakthroughs to functional implementations. The review reveals groundbreaking innovations in heat manipulation through the exploration of both classical and non-classical transport regimes, topological thermal control mechanisms, and quantum-informed phonon engineering strategies. By bridging physical insights like non-Hermitian thermal dynamics and valleytronic phonon transport with cutting-edge applications, we demonstrate paradigm-shifting capabilities: environment-adaptive thermal cloaks, AI-optimized metamaterials, and nonlinear thermal circuits enabling heat-based computation. Experimental milestones include 3D thermal null media with reconfigurable invisibility and thermal designs breaking classical conductivity limits. This collaborative effort establishes an indispensable roadmap for physicists, highlighting pathways to quantum thermal management, entropy-controlled energy systems, and topological devices. As thermal metamaterials transition from laboratory marvels to technological cornerstones, this work provides the foundational lexicon and design principles for the coming era of intelligent thermal matter.

We present a general framework for determining the power-efficiency trade-off relations across arbitrary thermal machines, addressing the lack of unified optimization results stemming from their diverse functionalities (e.g., heat engines, refrigerators, and heat pumps). For time-dependent cycle irreversibility A(τ) following a τ^-α power law, where α is an interaction-dependent parameter, we show that engineering the interactions between thermal machines and reservoirs enables control over the trade-off relations, with the efficiency at maximum power approaching Carnot efficiency as α increases. Setting α=1 naturally recovers typical low-dissipation regime results. Additionally, we derive the first power-efficiency trade-off for finite-time quantum adiabatic Otto machines with τ^-2-scaling. This work establishes a unified constraint for thermodynamic cycles across non-equilibrium regimes, facilitating consistent optimization of diverse thermal devices in practice.

Curzon and Ahlborn’s 1975 paper, a pioneering work that inspired the birth of the field of finite-time thermodynamics, unveiled the efficiency at maximum power (EMP) of the endoreversible Carnot heat engine, now commonly referred to as the Curzon–Ahlborn (CA) engine. Historically, despite the significance of the CA engine, similar findings had emerged at an earlier time, such as the Yvon engine proposed by J. Yvon in 1955 that shares the exact same EMP, that is, the CA efficiency ηCA. However, the special setup of the Yvon engine has circumscribed its broader influence. This paper extends the Yvon engine model to achieve a level of generality comparable to that of the CA engine. With the power expression of the extended Yvon engine, we directly explain the universality that ηCA is independent of the heat transfer coefficients between the working substance and the heat reservoirs. A rigorous comparison reveals that the extended Yvon engine and CA engine represent the steady-state and cyclic forms of the endoreversible Carnot heat engine, respectively, and are equivalent.

Carnot efficiency sets a fundamental upper bound on the heat engine efficiency, attainable in the quasi-static limit, albeit at the cost of completely sacrificing power output. In this Letter, we present a minimal heat engine model that can attain Carnot efficiency while achieving maximum power output. We unveil the potential of intrinsic divergent physical quantities within the working substance, such as degeneracy, as promising thermodynamic resources to break through the universal power-efficiency trade-off imposed by nonequilibrium thermodynamics for conventional heat engines. Our findings provide novel insights into the collective advantage in harnessing energy of many-body interacting systems.

The coercivity panorama for characterizing the dynamic hysteresis in interacting systems across multiple timescales is proposed by Chen et al. in a companion paper. For the stochastic ϕ^4 model under periodic driving of rate v_H, the coercivity landscape H_c(v_H) exhibits plateau features at a characteristic rate v_P with the corresponding coercivity H_P. Below this plateau (v_Hv_P), scaling in the fast-driving regime, H_c∼ v_H^1/2, is completely different from that, H_c-H_P∼ (v_H-v_P)^2/3, in the post-plateau slow-driving regime. The emergence of the plateau with a finite-size scaling reflects the competition between the thermodynamic limit and the quasi-static limit. In this paper, we provide detailed analytical proofs and numerical evidence supporting these results. Moreover, to demonstrate the coercivity panorama in concrete physical systems, we study the magnetic hysteresis in the Curie-Weiss model and analyze its finite-size effects. We reveal that finite-time coercivity scaling shows model-specific behavior only in the fast-driving regime, while exhibiting universal characteristics elsewhere.

One famous paper in this field [Am. J. Phys. 43, 22 (1975)] unveiled the efficiency at maximum power (EMP) of the endo-reversible Carnot heat engine, now known as the Curzon-Ahlborn (CA) engine, making a pioneering contribution to the genesis of finite-time thermodynamics. Despite its significance, similar findings have surfaced throughout history; for instance, the Yvon engine proposed by J. Yvon in 1955 shares the exact same EMP as the CA engine. This study extends Yvon's original approach to reanalyze the finite-time optimization of the CA engine. Our investigation not only bridges the gap between the Yvon engine and the CA engine but also serves as a suitable example for teaching undergraduate thermodynamics and engineering thermodynamics, given its concise and easy-to-understand derivation.

We propose a finite-time quantum Szilard engine (QSE) with a quantum particle with spin as the working substance (WS) to accelerate the operation of information engines. We introduce a Maxwell's demon (MD) to probe the spin state within a finite measurement time $t_{{\rm M}}$ to capture the which-way information of the particle, quantified by the mutual information $I(t_{\rm{M}})$ between WS and MD. We establish that the efficiency $\eta$ of QSE is bounded by $\eta\leq1-(1-\eta_{\rm{C}}){\rm ln}2/I(t_{{\rm M}})$, where $I(t_{{\rm M}})/\rm{ln}2$ characterizes the ideality of quantum measurement, and approaches $1$ for the Carnot efficiency reached under ideal measurement in quasi-static regime. We find that the power of QSE scales as $P\propto t_{{\rm M}}^{3}$ in the short-time regime and as $P\propto t_{\rm M}^{-1}$ in the long-time regime. Additionally, considering the energy cost for erasing the MD's memory required by Landauer's principle, there exists a threshold time that guarantees QSE to output positive work.

Microscopic particle separation plays vital role in various scientific and industrial domains. In this Letter, we propose a universal non-equilibrium thermodynamic approach, employing the concept of Shortcuts to Isothermality, to realize controllable separation of overdamped Brownian particles. By utilizing a designed ratchet potential with temporal period $\tau$, we find in the slow-driving regime that the average particle velocity $\Bar{v}_s\propto\left(1-D/D^*\right)\tau^{-1}$, indicating that particles with different diffusion coefficients $D$ can be guided to move in distinct directions with a preset $D^*$. Furthermore, we reveal that there exists an extra energetic cost with a lower bound $W_{\rm{ex}}^{(\rm{min})}\propto\mathcal{L}^{2}\Bar{v}_s$, alongside a quasi-static work consumption. Here, $\mathcal{L}$ is the thermodynamic length of the driving loop in the parametric space. We numerically validate our theoretical findings and illustrate the optimal separation protocol (associated with $W_{\rm{ex}}^{(\rm{min})}$) with a sawtooth potential. This study establishes a bridge between thermodynamic process engineering and particle separation, paving the way for further explorations of thermodynamic constrains and optimal control in ratchet-based particle separation.

In this paper, we summarize the historical development of finite-time thermodynamics and review the current state of research over the past two decades in this field, focusing on fundamental constraints of finite-time thermodynamic cycles, optimal control and optimization of thermodynamic processes, the operation of unconventional heat engines, and experimental progress.

We study temperature fluctuations in mesoscopic $N$-body systems undergoing non-equilibrium processes from the perspective of stochastic thermodynamics. By introducing a stochastic differential equation, we describe the evolution of the system's temperature during an isothermal process, with the noise term accounting for finite-size effects arising from random energy transfer between the system and the reservoir. Our analysis reveals that these fluctuations make the extensive quantities (in the thermodynamic limit) deviate from being extensive for consistency with the theory of equilibrium fluctuation. Moreover, we derive finite-size corrections to the Jarzynski equality, providing insights into how heat capacity influences such corrections. Also, our results indicate a possible violation of the principle of maximum work by an amount proportional to $N^{-1}$. Additionally, we examine the impact of temperature fluctuations in a finite-size quasi-static Carnot engine. We show that irreversible entropy production resulting from the temperature fluctuations of the working substance diminishes the average efficiency of the cycle as $\eta_{\rm{C}}-\left\langle \eta\right\rangle \sim N^{-1}$, highlighting the unattainability of the Carnot efficiency $\eta_{\rm{C}}$ for mesoscopic-scale heat engines even under the quasi-static limit

Saxon bowl is a timing device with a hole in the bottom. In this paper, a physical model of cylindrical Saxon bowl sinking in water is established. First, this paper determined the critical conditions for the successful sinking of the Saxon bowl, including the conditions under which water can flood into the bowl and the conditions under which the bowl is initially released. Then, this paper studied the dynamics of the system after the water pouring into the bowl and deduced the quantitative relationship between the sinking time of the bowl and the geometric characteristics of the bowl. Furthermore, experiments are designed to test the correctness of the theoretical model. Compared with the experimental sinking time of the bowl, the corresponding theoretical value is generally smaller. After taking into account the viscous resistance and differential pressure resistance in the process of bowl sinking, the loss of submerged jet and head at the orifice, and the influence of surface tension before the bowl is completely immersed, the consistency between theory and experiment is increased obviously. Finally, a wall function is introduced to analyze the sinking time of various shapes of Saxon bowls.

In most thermodynamics textbooks, polytropic processes are introduced by defining the process equation rather than explaining their physical origin. In this paper, we propose a simple model to realize a polytropic process for an ideal gas. The ideal gas is compressed or expanded quasi-statically while in thermal contact with a finite reservoir with constant heat capacity, and the entire system is thermally isolated from the outside environment. We propose an experimental implementation of our model with realistic parameters.