Transient Thermodynamic Backflow

• Transient Thermodynamic Backflow is the phenomenon where energy, heat, or information momentarily reverses its typical flow direction due to memory effects and non-Markovian dynamics.
• It spans quantum systems, classical thermal circuits, and hydrodynamic conductors, each exhibiting unique signatures like temperature undershoots and oscillatory heat pulses.
• Researchers use models such as the spin–boson model and thermal inductor analogues to quantify, control, and harness backflow for enhanced device performance and energy management.

Transient thermodynamic backflow refers to the phenomenon—in both quantum and classical nonequilibrium systems—by which energy or heat, probability density, or even information, transiently flows in the reverse direction of a system's typical thermodynamic arrow. In the thermodynamic context, it most commonly denotes episodes where, due to memory effects, dynamical correlations, or non-diffusive transport, heat “backflows” from colder to hotter subsystems, or dissipative channels inject energy into a system rather than removing it. Such reversals do not violate the second law, as they are strictly transient and constrained by the global statistical properties of the ensemble. This concept reveals deep links between non-Markovian dynamics, reservoir engineering, thermal circuit design, and the operational control of dissipation in quantum and classical devices.

1. Foundational Models and Definitions

The canonical setting for microscopic heat backflow is the spin–boson model—a two-level system (TLS) linearly coupled to a bosonic bath with Ohmic spectral density. The total Hamiltonian is

$$H = -\frac{\hbar\omega}{2}\sigma_x + \sum_k\hbar\omega_k b_k^\dagger b_k + \sigma_z\sum_k c_k(b_k^\dagger + b_k).$$

The bath spectral density is taken as $$J(\omega) = \hbar\gamma\omega/[1+(\omega/\omega_c)^2]^2$$, with cutoff $\omega_c \gg \omega$, ensuring non-Markovianity is inherited from low temperature ($T$) and strong coupling ($\gamma$), not spectral shaping (Schmidt et al., 2016).

In the stochastic Liouville–von Neumann (SLN) framework, the instantaneous heat flux operator is given by

$$\hat{J}(t) = \frac{dH_r(t)}{dt} = \frac{i}{\hbar}[H, H_r](t)$$

and, after reduction, the mean heat current is

$$j_Q(t) = -\omega \langle \xi(t) \sigma_y(t) \rangle,$$

where $\xi(t)$ encodes the colored (non-Markovian) quantum noise from the bath.

Criterion for backflow: In the convention where $j_Q < 0$ is net energy from system to bath, transient thermodynamic backflow corresponds to $j_Q(t) > 0$. Structurally, the time-local force-force correlation (bath memory kernel)

$$K(t) = \mathrm{Re}C(t) = \int_{0}^{\infty} \frac{d\omega}{\pi} J(\omega)\coth(\beta\hbar\omega/2)\cos\omega t$$

can attain negative values, which physically means the reservoir “remembers” an excitation and can transiently pump energy back into the TLS.

2. Transient Backflow Mechanisms: Quantum, Classical, and Hydrodynamic

Quantum Regimes

Non-Markovian quantum systems such as the spin–boson model, a Lindblad-driven qubit with coherent catalyst coupling, or wave-packet propagation in dissipative environments exhibit thermodynamic backflow when the environmental memory kernel enables a reversal in the direction of net heat or probability flow. As analyzed in (Schmidt et al., 2016) and (Zhao, 7 Dec 2025), quantum backflow is not a steady-state feature but appears as a brief pulse after non-equilibrium preparation, exhibit temporal structure sensitive to temperature, coupling strength, and external drive amplitude. In (Zhao, 7 Dec 2025), a qubit (quantum battery) coupled to a harmonic oscillator catalyst displays heat current $J(t) < 0$ (backflow into the battery) over timescales $t \sim O(1/g)$, directly compensating for dephasing losses and enabling enhanced extractable work (ergotropy).

Inertial and Circuit-based Thermal Backflow

Thermodynamic backflow also manifests in classical circuits equipped with elements that introduce inertia—the “thermal inductor” paradigm. Adding a Peltier element in series with an inductor enables heat current reversal by imposing an $L(dJ/dt)$ term in the heat transport equation (thermal LCR analogy). When the circuit is activated, the inductance resists rapid changes, so the direction of thermal current can overshoot, resulting in heat flowing from cold to hot, before relaxing in a damped oscillatory manner. This effect was directly seen in experiments, where a thermally coupled copper block undershot the reservoir temperature by up to 2.7 K without any external driving (Schilling et al., 2018).

Hydrodynamic and Viscous Backflow

In ultra-clean crystals (graphite, h-BN), collective phonon motion and a finite “thermal viscosity” $\eta_T$ render heat transport hydrodynamic. The fourth-order term in the viscous heat equation

$$\rho c_p \partial_t T - \nabla \cdot (\kappa \nabla T) + \nabla \cdot [\eta_T \nabla^2(\nabla T)] = 0$$

permits wave-like solutions. When the device is driven at or near a thermal resonance, backflow arises as local reversals of the temperature gradient and heat flux, with amplitudes $0.1\mbox{–}1\,\mathrm{K}$ and ns-scale lifetimes, directly measurable via time-resolved thermoreflectance (Dragašević et al., 2023).

3. Quantification, Signatures, and Theoretical Formulation

The quantifiers of transient thermodynamic backflow depend on the context:

• Energy/Heat Flux: For a quantum system, integrated positive-area of $j_Q(t)$:

$H_+(T, \gamma) = \int_{j_Q>0} j_Q(t)\, dt$

provides the overall energetic weight of backflow pulses (Schmidt et al., 2016).

• Probability Current (Quantum Backflow): For probability density,

$F(\tau) = -\int_0^\tau j(0, t) dt$

with $F(\tau) > 0$ indicating net backflow.

• Thermodynamic Force–Flow Structure: The quantum affinity operator

$A(t) = \ln\rho_s^\beta - \ln\rho_s(t)$

acts as the driving "force" whose rate of change predicts the direction of flow or backflow (i.e., entropy production rate strictly positive unless the process is strictly reversible), with transient increases in mean affinity marking information or energy backflow (Ahmadi et al., 2018).

• Experimental Signatures: Inmesoscopic circuits and hydrodynamic crystals, direct observable effects include temperature undershoots (below the ambient bath), sign-reversals in heat flux, and resonance-amplified thermal oscillations.

4. Physical Origins and Dependences

Quantum regime: At low $T$ and strong $\gamma$, bath memory kernels become negative over finite intervals, maximizing heat backflow. Non-Markovian environments, strong system-bath coupling, and resonant driving further amplify the magnitude and duration of the effect. Individually, lowering $T$ initiates earlier and stronger backflow pulses. Increasing $\gamma$ deepens negative memory kernel excursions but blurs revival structure.

Classical regime: In thermal circuits, optimal backflow occurs when the inductor and resistor values are matched to the characteristic heat capacity and conductance ($L \approx RC/k$). The resulting thermal oscillation causes a single transient reversal before the system equilibrates.

Hydrodynamic heat transport: Finite $\eta_T$ (phononally viscous) materials sustain transient thermal waves and local gradient reversal. Backflow is controlled by the magnitude of $\eta_T$, device size, boundary slip/reflectivity, and proximity to resonance.

External driving: Coherent periodic fields or boundary pulses can either suppress or enhance thermodynamic backflow depending on phase and resonance conditions.

5. Contrasts with Information Backflow and Other Modalities

A key conclusion from (Schmidt et al., 2016) is that heat (energy) backflow and information backflow are distinct phenomena: revivals in the trace-distance (Breuer-Jane-Luo-Piilo) measure of distinguishability between quantum states do not, in general, coincide temporally with energy backflow events. Observable projections relevant for energy (e.g., $\langle \xi(t)\sigma_y(t) \rangle$) and for information (state distinguishability) are sensitive to different aspects of system–bath correlations.

In open quantum systems, non-Markovian memory can drive both transient heat and information backflow, but the physical mechanisms and control strategies (drive parameters, bath engineering) for one may not optimize the other. This distinction means that applications targeting information preservation (e.g., error correction) versus energy injection (e.g., catalyzed quantum batteries) may require different protocols.

6. Applications and Control in Engineered Systems

Quantum technologies: Reservoir engineering (manipulation of bath spectral properties, tailored driving) allows for temporally precise heat backflow, enabling:

• Short-time cooling “kicks” in quantum refrigerators.
• Maxwell-demon–type feedback to extract work during backflow windows.
• Enhanced quantum battery performance by transiently creating non-passive, high-ergotropy states (Zhao, 7 Dec 2025).

Thermal circuits: The “thermal inductor” effect allows passive cooling of objects below ambient, pulse-driven thermal oscillators and logic, and energy harvesting machines that transiently violate the directionality predicted by Fourier’s law but always respect the second law (Schilling et al., 2018).

Hydrodynamic conductors: Device-boundary engineering (e.g., slip conditions, resonance tuning) in mesoscopic materials enables amplification and control of transient thermal waves and vortexed heat flows for advanced phononic/electronic applications (Dragašević et al., 2023).

Fluidic systems: In convective flows (e.g., thermosyphons, U-bend pipes), backflow episodes correspond to large-scale flow reversals or secondary vortex inversions. Their prediction and control exploit both low-dimensional modal precursors (captured via DMD) and advanced data assimilation strategies (Reagan et al., 2015, Skillen et al., 2019).

7. Entropy Production, the Second Law, and Physical Consistency

All experimentally and theoretically observed transient backflow phenomena rigorously comply with the second law of thermodynamics. While momentary reversals in heat or probability current can occur, the total entropy production rate remains positive when integrated over the full process. In circuit models, the only strictly dissipative terms are Ohmic (Joule) heating and steady-state Fourier (conductive) heat leak, while the “inertial” or “backflow” contributions store, but do not dissipate, energy (Schilling et al., 2018). In the quantum setting, the non-monotonicity of the quantum affinity during backflow windows transiently reverses the arrow of thermodynamic force, but global irreversibility and energetic ordering are restored at longer timescales (Ahmadi et al., 2018).

8. Summary Table: Principal Regimes and Mechanisms

Physical Regime	Backflow Mechanism	Key Signatures / Control
Quantum open system (spin-boson)	Negative bath memory kernel $K(t)$ drives $j_Q(t)>0$	Non-Markovianity, strong coupling, low $T$
Quantum battery with catalyst	Coherent energy return via system-catalyst exchange	Ergotropy growth, drive/coupling optimization
Classical thermal circuit	Thermal inertia (inductor) delays relaxation, reverses current	Damped oscillatory $\Delta T(t)$ undershoot
Hydrodynamic conductor	Viscous heat flux, device resonance amplifies wave backflow	MHz-scale $\Delta T$ oscillations
Turbulent convection, flow loop	Baroclinic vorticity, inertia/thermal stratification	Flow reversal, large-scale vortex switching

Each regime realizes transient thermodynamic backflow through a combination of memory, inertia, nonlinearity, or resonance, with timescales and observability set by the interplay of dissipation, drive amplitude, system-bath coupling, and boundary conditions. All forms of backflow remain fundamentally constrained by the second law of thermodynamics and are integral to advanced control strategies in thermodynamic devices, quantum information, and nonequilibrium matter (Schmidt et al., 2016, Schilling et al., 2018, Dragašević et al., 2023, Zhao, 7 Dec 2025, Ahmadi et al., 2018).