Instantaneous Excess Work in Nonequilibrium Processes

• Instantaneous Excess Work is defined as the extra energy cost generated when a system is rapidly driven out of equilibrium through sudden changes in its control parameters.
• It is quantified by decomposing total work into reversible and irreversible components, using methods such as two-point measurement schemes and geometric line integrals in nonequilibrium steady states.
• The concept applies across classical, quantum, and relativistic systems, providing insights into dissipation, entropy production, and critical behavior in fast, nonequilibrium processes.

Instantaneous excess work (also termed instantaneous irreversible work or simply excess work) quantifies the additional energetic cost that arises when a physical system—classical or quantum—is driven out of equilibrium by rapid (including strictly instantaneous) changes in its control parameters. This notion generalizes the concept of dissipation to arbitrary nonequilibrium processes, providing a rigorous basis for energy spreads, entropy production, and deviations from reversibility in a wide range of settings, from macroscopic thermodynamics to quantum many-body systems and relativistic quantum engines.

1. General Framework for Instantaneous Excess Work

In generalized nonequilibrium thermodynamics, the total infinitesimal work $dW(t)$ is decomposed into reversible and irreversible components,

$dW(t) = d_{e}W(t) + d_{i}W(t),$

where $d_{e}W(t)$ is the exchange (reversible) work performed against the external fields (such as pressure or chemical potential of the medium), and $d_{i}W(t)$ is the instantaneous excess (irreversible) work generated internally through dissipation during deviations from equilibrium. For example, in pressure–volume work,

$$d_{e}W(t) = P_{0}\,dV(t), \quad d_{i}W(t) = [P(t) - P_{0}]\, dV(t),$$

where $P_{0}$ is the external pressure and $P(t)$ the instantaneous internal pressure of the system. The excess work $d_{i}W(t)$ directly measures the energy dissipated via non-equilibrium field mismatches and, thermodynamically, this contribution is always non-negative and coincides with the increase of internal heat via $d_{i}Q(t) = d_{i}W(t)$, ensuring consistency with the second law and entropy production (Gujrati, 2011).

2. Quantum Instantaneous Excess Work: Measurement and Statistics

In quantum systems, instantaneous excess work is operationally defined via the two-point measurement scheme: the system, initially in state $\rho_i$ and Hamiltonian $H(\lambda_i)$, undergoes a sudden quench of the parameter $\lambda_i \to \lambda_f$. The average work is

$$\langle W \rangle = \text{Tr}\left[ H(\lambda_f)\, \rho_i \right] - \text{Tr}\left[ H(\lambda_i)\, \rho_i \right].$$

The excess (irreversible) work is then identified as

$$W_{\mathrm{irr}} = \langle W \rangle - \left[ E_0(\lambda_f) - E_0(\lambda_i) \right],$$

where $E_0(\lambda)$ is the ground state (or equilibrium free energy, for general states), connecting the nonequilibrium and adiabatic reference energies. In small quenches, the expansion

$$W_{\mathrm{irr}} \approx -\frac{1}{2} (\delta\lambda)^2 \left. \frac{\partial^2 E_0}{\partial\lambda^2} \right|_{\lambda_i},$$

exhibits how the excess work reflects system susceptibilities and critical behavior (Mascarenhas et al., 2013, Jr et al., 2015, Nazé, 2023).

3. Excess Work in Nonequilibrium Steady States and its Geometric Structure

For transitions between nonequilibrium steady states (NESS), instantaneous excess work is isolated from the divergent "housekeeping" work (needed to maintain the NESS) by subtracting the steady power flow,

$$W_{\text{ex}} = \int_{t_0}^{t_f} \left[ J^W(\lambda(t), t) - J^W_{\text{ss}}(\lambda(t)) \right]\,dt,$$

where $J^W$ is the instantaneous input power and $J^W_{\text{ss}}$ is the power maintaining the steady state. Under quasistatic and linear response conditions, the excess work reduces to a geometric line integral in control parameter space,

$W_{\text{ex}} = \int_{C} A_i(\lambda) d\lambda_i,$

with $A_i(\lambda)$ the vector potential determined by response functions—an intrinsic measure of path-dependence for nonequilibrium thermodynamics (Yuge, 2013).

Table: Key Excess Work Quantities in NESS Transitions

Quantity	Definition	Physical Role
$W_{\text{tot}}$	$\int J^W(\lambda(t), t) dt$	Total input work
$W_{\text{hk}}$	$\int J^W_{\text{ss}}(\lambda(t)) dt$	Housekeeping (maintenance) work
$W_{\text{ex}}$	$W_{\text{tot}} - W_{\text{hk}}$	Excess work (intrinsic cost)
$A_i(\lambda)$	Path-integral vector potential from response functions	Geometric structure for $W_{\text{ex}}$

In these protocols, the instantaneous excess power $P_{\text{ex}}(t)$ is given by $A_i(\lambda(t))\dot\lambda_i(t)$, which is experimentally accessible once the response function is known.

4. Instantaneous Quenches in Quantum Field Theory and Many-Body Systems

Sudden (instantaneous) quenches lead to nontrivial excess energy injection, especially in quantum field theory and interacting models. For a free scalar field, the excess energy density following a mass quench $m_{\text{in}} \to m_{\text{out}}$ is

$$\Delta \mathcal{E}^{\text{inst}} = \mathcal{E}_{\text{quench}} - \mathcal{E}_{\text{ground}},$$

where divergences appear for spatial dimensions $d \geq 4$. The divergence structure,

$$\Delta\mathcal{E}^{\text{inst}} \sim m^4\,\Lambda^{d-4},$$

indicates that truly instantaneous quenches are ill-defined for relevant operators with scaling dimension $\Delta > d/2$. In lower dimensions $(d < 4)$, the excess energy is finite and matches the limit of fast, but smooth, quenches (Das et al., 2015).

For quantum spin chains (e.g., the anisotropic XY or Ising models), the exact analytical form of the instantaneous excess work and its fluctuations can be computed. The approach involves the full statistics of work via characteristic functions and connects critical behavior (such as quantum phase transitions) to non-analyticities in the excess work (Jr et al., 2015).

5. Cumulant Expansion and Nonlinear Response

The excess work can be systematically expanded via cumulant series in the parameter change $\Delta\lambda$,

$$W_{\mathrm{irr}} = \sum_{n=2}^{\infty} \frac{(-\Delta\lambda)^n}{n!}\,\kappa_n,$$

where $\kappa_n$ are the $n$-th order cumulants of the generalized conjugate force. For an instantaneous quench, these reduce to equal-time equilibrium cumulants (or moments) and guarantee the non-negativity of excess work at each order, consistent with the second law. The expansion provides a well-controlled nonlinear response analysis for irreversible work beyond the linear regime (Nazé, 2023).

6. Instantaneous Excess Work in Counterdiabatic and Engine Protocols

In shortcuts-to-adiabaticity, particularly counterdiabatic (CD) driving, quantifying the energetic cost via instantaneous excess work provides a direct bridge to quantum speed limits (Mandelstam–Tamm bound). For Hamiltonian protocols with time-dependent control, the excess work,

$$W_{\text{ex}}(t) = \text{Tr}\big([ \rho(t) - \rho_{\text{ad}}(t) ] H(t) \big)$$

vanishes under perfect CD protocols unless reinterpretation of protocol duration is applied. By freezing the scaling of auxiliary controls, a nonzero $W_{\text{ex}}$ is obtained, revealing an intrinsic energetic cost for finite-time adiabaticity—a physical manifestation of energy delocalization in the instantaneous spectrum (Kamizaki et al., 2 Dec 2025).

Similarly, in relativistic quantum Otto engines with instantaneous (delta-switching) system-bath couplings, the total extracted work per engine cycle for sharply separated isochores is

$W_{\mathrm{inst}} = \Delta\Omega\,(p_1 - p),$

where $p_1$ is the excited state occupation after the first delta-kick and $p$ is initial occupation. The excess work relative to the quasistatic limit, $$W_{\mathrm{ex}} = W_{\mathrm{inst}} - W_{\mathrm{ad}}$$, captures the genuinely nonequilibrium contribution due to field-mediated signaling and the instantaneous nature of the interactions (Gallock-Yoshimura, 2023).

7. Physical Interpretation, Critical Signatures, and Observability

Instantaneous excess work serves as a unifying metric of irreversibility, energetic delocalization, and entropy production in driven—particularly sudden—processes. Its value is path-dependent in far-from-equilibrium protocols and identifies criticality in many-body systems via nonanalyticities and singularities (e.g., close to quantum critical points, excess work shows discontinuities or divergences in its field or parameter derivatives) (Mascarenhas et al., 2013, Jr et al., 2015). Furthermore, in both experimental and theoretical settings, instantaneous excess work is accessible via measurement statistics, direct evaluation of fluctuation cumulants, or, in quantum technologies, via population and correlation measurements of work distribution and engine performance.

The breadth of applicability—from classical nonequilibrium thermodynamics (Gujrati, 2011), steady-state transitions (Yuge, 2013), and the full quantum regime including quantum information processing (Kamizaki et al., 2 Dec 2025, Gallock-Yoshimura, 2023)—demonstrates the centrality of instantaneous excess work as a quantitative and operational cornerstone for understanding irreversibility across physical sciences.