Semi-Quenched Entropy: An Intermediate Regime

• Semi-quenched entropy is an intermediate entropy concept that splits the total entropy into a static environment cost and a dynamic Markov cost, bridging annealed and quenched regimes.
• It is characterized by an interpolation parameter (α) that effectively modulates the penalty of environment fluctuations in models like random walks in dynamic random environments.
• The concept finds broad applications from large deviations and disordered quantum criticality to holographic semi-local liquids and weak quenches in thermally isolated systems.

Semi-quenched entropy denotes an intermediate entropy concept between fully averaged or annealed descriptions and fully quenched descriptions. The term is not introduced as a formal definition in the cited works, but the underlying structure is explicit in several settings. In the most precise formulation, developed for random walk in a dynamic random environment, the averaged level-3 entropy splits into a static environment cost and a dynamic Markov cost; this decomposition naturally suggests entropy functionals that are neither purely annealed nor purely quenched (Rassoul-Agha et al., 2016). In other contexts—quenched disordered entanglement, semi-local holography, and thermally isolated quenches—the same phrase is most naturally interpreted as describing a regime in which some degrees of freedom are frozen or typical while others are integrated out, dynamically tilted, or softly penalized.

1. Conceptual core

The clearest mathematical source for the notion is the entropy decomposition

$$h(\mu\,|\,P_0)=h_{\mathcal{S}_{0,\infty}}(\mu_\Omega\,|\,\mathbb{P})+H_q(\mu).$$

Here $h(\mu|P_0)$ is the averaged specific relative entropy density of the empirical process, $h_{\mathcal{S}_{0,\infty}}(\mu_\Omega|\mathbb{P})$ is the specific relative entropy rate of the environment marginal relative to the environment law, and $H_q(\mu)$ is a Donsker–Varadhan-type relative entropy for the environment Markov process. In the terminology suggested by this decomposition, $h_{\mathcal{S}_{0,\infty}}$ is a static environment cost and $H_q$ is a dynamic Markov cost (Rassoul-Agha et al., 2016).

This yields a precise distinction between three regimes. In the fully averaged case, one pays both the cost of reshaping the environment law and the cost of changing the dynamics. In the fully quenched case, the environment is frozen in the sense that one minimizes only the dynamic entropy, subject to absolute continuity and lower semicontinuous regularization. A semi-quenched functional is then naturally interpreted as one that retains the Markov entropy $H_q(\mu)$ while allowing a controlled environment cost rather than eliminating it completely.

A natural interpolation suggested by this structure is

$$I_{3,\alpha}(\mu):=\alpha\,h_{\mathcal{S}_{0,\infty}}(\mu_\Omega|\mathbb{P})+H_q(\mu),\qquad \alpha\in[0,1],$$

assuming $\mu$ is $S$-invariant and $h(\mu|P_0)$0, and $h(\mu|P_0)$1 otherwise. In this interpretation, $h(\mu|P_0)$2 gives the averaged rate, $h(\mu|P_0)$3 is closely related to the quenched rate after lower semicontinuous regularization and absolute continuity constraints, and $h(\mu|P_0)$4 describes an intermediate regime in which environment fluctuations are penalized but not fully suppressed (Rassoul-Agha et al., 2016).

2. Random walks in dynamic random environments

The model in which the semi-quenched structure is sharpest is a random walk with bounded jumps on $h(\mu|P_0)$5 in a temporally i.i.d. and spatially translation-invariant dynamic random environment. The allowed one-step increments form a finite set $h(\mu|P_0)$6, and the environment is a collection

$h(\mu|P_0)$7

where each $h(\mu|P_0)$8 is a probability measure on $h(\mu|P_0)$9. Given $h_{\mathcal{S}_{0,\infty}}(\mu_\Omega|\mathbb{P})$0, the quenched walk is a time-inhomogeneous Markov chain, while under the averaged law the walk is a homogeneous Markov chain with jump kernel $h_{\mathcal{S}_{0,\infty}}(\mu_\Omega|\mathbb{P})$1 (Rassoul-Agha et al., 2016).

The central object is the environment as seen from the particle,

$h_{\mathcal{S}_{0,\infty}}(\mu_\Omega|\mathbb{P})$2

which is a Markov chain on $h_{\mathcal{S}_{0,\infty}}(\mu_\Omega|\mathbb{P})$3. Large deviations are formulated for the empirical process

$h_{\mathcal{S}_{0,\infty}}(\mu_\Omega|\mathbb{P})$4

a level-3 object recording the entire process of environment-step pairs seen along the walk. In the averaged setting, the level-3 rate is

$h_{\mathcal{S}_{0,\infty}}(\mu_\Omega|\mathbb{P})$5

where $h_{\mathcal{S}_{0,\infty}}(\mu_\Omega|\mathbb{P})$6 is a specific relative entropy density. In the quenched setting, under the moment assumption

$h_{\mathcal{S}_{0,\infty}}(\mu_\Omega|\mathbb{P})$7

the level-3 rate is

$h_{\mathcal{S}_{0,\infty}}(\mu_\Omega|\mathbb{P})$8

with $h_{\mathcal{S}_{0,\infty}}(\mu_\Omega|\mathbb{P})$9 defined from the relative entropy of the actual transition kernel under $H_q(\mu)$0 relative to the underlying kernel $H_q(\mu)$1, averaged over the stationary past (Rassoul-Agha et al., 2016).

The semi-quenched interpretation enters because these two rate functions are not unrelated alternatives. They are connected by the exact decomposition above, and the comparison chain

$H_q(\mu)$2

shows that averaged and quenched descriptions differ precisely by the treatment of the environment marginal and the lower semicontinuous envelope. In this setting, semi-quenched entropy is not a metaphor but a direct reading of the rate-function geometry (Rassoul-Agha et al., 2016).

3. Variational structure, contractions, and minimizers

The level-3 theory contracts to a level-1 large deviation principle for the velocity $H_q(\mu)$3. In the averaged case,

$H_q(\mu)$4

and the classical Cramér formula is

$H_q(\mu)$5

For $H_q(\mu)$6, the minimizer is the unique $H_q(\mu)$7-invariant measure $H_q(\mu)$8, defined through exponential tilting. Under $H_q(\mu)$9, the slabs $h_{\mathcal{S}_{0,\infty}}$0 are i.i.d., the quenched walk under $h_{\mathcal{S}_{0,\infty}}$1 is Markov, and the effective kernel satisfies

$h_{\mathcal{S}_{0,\infty}}$2

$h_{\mathcal{S}_{0,\infty}}$3-a.s. (Rassoul-Agha et al., 2016).

When $h_{\mathcal{S}_{0,\infty}}$4 on $h_{\mathcal{S}_{0,\infty}}$5, this tilting admits a Doob $h_{\mathcal{S}_{0,\infty}}$6-transform representation: $h_{\mathcal{S}_{0,\infty}}$7 for some $h_{\mathcal{S}_{0,\infty}}$8, $h_{\mathcal{S}_{0,\infty}}$9 a.s. The quenched contraction is

$H_q$0

and, for $H_q$1,

$H_q$2

This is the precise point at which the static term is dropped and only the dynamic entropy remains (Rassoul-Agha et al., 2016).

The equivalence theorem for $H_q$3 identifies when the averaged and quenched descriptions coincide: $H_q$4 Thus the averaged minimizer is also the quenched minimizer exactly when the environment marginal is typical and incurs no static cost. This is the canonical semi-quenched criterion: all cost lies in dynamically re-encoding the walk, while the environment law itself remains typical. When this fails, the quenched minimizer is a different process. The paper makes this explicit in the spatially constant environment example, where $H_q$5, the quenched rate blows up at $H_q$6, and a different Markov process $H_q$7 with $H_q$8 realizes the quenched rate (Rassoul-Agha et al., 2016).

4. Entanglement in quenched disordered quantum criticality

A different but structurally related usage arises in quenched disordered entanglement. For a $H_q$9D Dirac fermion in a static random magnetic field,

$H_q(\mu)$0

with Gaussian disorder

$H_q(\mu)$1

the disorder can be Hodge-decomposed and absorbed by an axial gauge transformation. The resulting disorder dressing appears as vertex operators of a scalar field $H_q(\mu)$2, and the interacting $H_q(\mu)$3D Green’s function after disorder average becomes

$H_q(\mu)$4

or, in momentum space,

$H_q(\mu)$5

The dimensional-reduction construction then lifts this lower-dimensional disorder-dressed propagator back to the $H_q(\mu)$6D entanglement problem (Tang et al., 2022).

The entanglement entropy is computed by the replica formula

$H_q(\mu)$7

with disorder averaging performed first at the level of correlation functions and effective Green’s functions. The paper states that there is no separate annealed entropy; the entropy considered is that of the ground state after disorder averaging at the level of correlators. This suggests a semi-quenched interpretation: the random field is quenched and static, but the actual entanglement calculation is performed for a deterministic effective theory in which disorder has already been integrated into anomalous propagators (Tang et al., 2022).

For the random gauge problem, the entanglement entropy satisfies an area law and the disorder modifies the area-law coefficient. The explicit result is

$H_q(\mu)$8

The subleading correction due to finite correlation length is a universal function of the correlation length and disorder strength, and the finite term

$H_q(\mu)$9

is negative and behaves as an RG monotone. In this context, semi-quenched entropy refers not to a variational large-deviation functional but to an intermediate operational regime: disorder is frozen in time, averaged over statistically, and encoded in lower-dimensional effective interactions before entanglement is computed (Tang et al., 2022).

5. Holographic semi-local quantum liquids

In holography, the phrase acquires a thermodynamic and geometric meaning. Semi-local quantum liquids are finite-density states whose deep IR geometry is conformal to

$$I_{3,\alpha}(\mu):=\alpha\,h_{\mathcal{S}_{0,\infty}}(\mu_\Omega|\mathbb{P})+H_q(\mu),\qquad \alpha\in[0,1],$$0

with metric

$$I_{3,\alpha}(\mu):=\alpha\,h_{\mathcal{S}_{0,\infty}}(\mu_\Omega|\mathbb{P})+H_q(\mu),\qquad \alpha\in[0,1],$$1

Only time and the radial coordinate scale; the spatial coordinates are spectators. The finite-temperature generalization has entropy density

$$I_{3,\alpha}(\mu):=\alpha\,h_{\mathcal{S}_{0,\infty}}(\mu_\Omega|\mathbb{P})+H_q(\mu),\qquad \alpha\in[0,1],$$2

so $$I_{3,\alpha}(\mu):=\alpha\,h_{\mathcal{S}_{0,\infty}}(\mu_\Omega|\mathbb{P})+H_q(\mu),\qquad \alpha\in[0,1],$$3 for any $$I_{3,\alpha}(\mu):=\alpha\,h_{\mathcal{S}_{0,\infty}}(\mu_\Omega|\mathbb{P})+H_q(\mu),\qquad \alpha\in[0,1],$$4. The ground state is therefore thermally quenched in the sense of having no residual extensive entropy density, even though the IR geometry retains an emergent $$I_{3,\alpha}(\mu):=\alpha\,h_{\mathcal{S}_{0,\infty}}(\mu_\Omega|\mathbb{P})+H_q(\mu),\qquad \alpha\in[0,1],$$5 factor and nontrivial quantum structure (Erdmenger et al., 2013).

The entanglement entropy is computed by the Ryu–Takayanagi prescription for strip, sphere, and annulus regions. For a strip in the pure IR semi-local geometry, the connected minimal surface exists only for one specific width,

$$I_{3,\alpha}(\mu):=\alpha\,h_{\mathcal{S}_{0,\infty}}(\mu_\Omega|\mathbb{P})+H_q(\mu),\qquad \alpha\in[0,1],$$6

In the UV-complete geometry this becomes

$$I_{3,\alpha}(\mu):=\alpha\,h_{\mathcal{S}_{0,\infty}}(\mu_\Omega|\mathbb{P})+H_q(\mu),\qquad \alpha\in[0,1],$$7

and there is a connected–disconnected transition: for small $$I_{3,\alpha}(\mu):=\alpha\,h_{\mathcal{S}_{0,\infty}}(\mu_\Omega|\mathbb{P})+H_q(\mu),\qquad \alpha\in[0,1],$$8, the connected surface dominates; as $$I_{3,\alpha}(\mu):=\alpha\,h_{\mathcal{S}_{0,\infty}}(\mu_\Omega|\mathbb{P})+H_q(\mu),\qquad \alpha\in[0,1],$$9 increases, the connected solution degenerates into disconnected slabs; for $\mu$0, only the disconnected solution exists. For the annulus, the maximum width approaches the same strip scale,

$\mu$1

at large radii (Erdmenger et al., 2013).

For spherical entangling regions, by contrast, there is no connected–disconnected competition and no phase transition. The leading IR contribution obeys an area law,

$\mu$2

with explicit asymptotics

$\mu$3

There is no logarithmic violation of the area law. This supports a semi-quenched interpretation in which thermal entropy is quenched, long-distance spatial entanglement is partially quenched, and temporal criticality remains unquenched because of the $\mu$4 factor (Erdmenger et al., 2013).

6. Thermally isolated quenches and weak-driving regimes

A third usage concerns nonequilibrium entropy production in thermally isolated Hamiltonian systems. For a system with Hamiltonian $\mu$5, microcanonical initial conditions, and a driving protocol $\mu$6, the paper compares three entropy definitions: Swendsen’s canonical entropy,

$\mu$7

Boltzmann entropy $\mu$8, and Gibbs volume entropy $\mu$9. The canonical entropy satisfies

$S$0

and the key finite-$S$1 theorem is

$S$2

where $S$3 is the mean work in the associated canonical reference ensemble. For a macroscopic system, $S$4 is of order $S$5, the difference between microcanonical and canonical work averages is of order $S$6, and higher corrections are subextensive. Hence the extensive part of the entropy change does not become negative (Seifert, 2019).

For finite systems and sufficiently weak driving, however, the mean entropy change can be negative. For an infinitesimal quench $S$7, the leading non-negative term scales as $S$8, while subextensive corrections scale as $S$9 and $h(\mu|P_0)$00. The paper concludes that there is generically a one-sided small range $h(\mu|P_0)$01 in which the mean canonical entropy change becomes negative, of order $h(\mu|P_0)$02. Refined microcanonical Crooks relations show analogous behavior for $h(\mu|P_0)$03 and $h(\mu|P_0)$04: extensive negative entropy production is excluded in large systems, but order-$h(\mu|P_0)$05 negative mean changes remain possible (Seifert, 2019).

The harmonic-oscillator examples make the point explicit. For an $h(\mu|P_0)$06-dimensional isotropic oscillator with a stiffness quench $h(\mu|P_0)$07, the work is

$h(\mu|P_0)$08

with $h(\mu|P_0)$09 distributed according to a symmetric Beta law. In this model,

$h(\mu|P_0)$10

and $h(\mu|P_0)$11 for any $h(\mu|P_0)$12, whereas $h(\mu|P_0)$13 for $h(\mu|P_0)$14 and any $h(\mu|P_0)$15, and also in a shrinking interval $h(\mu|P_0)$16 for any $h(\mu|P_0)$17. In a distinct $h(\mu|P_0)$18D oscillator-to-disc quench, both Gibbs and canonical mean entropy changes are negative. In this nonequilibrium setting, semi-quenched entropy is most naturally interpreted as a weak-quench or partial-quench regime in which macroscopic entropy production remains non-negative while subextensive corrections can dominate the mean sign (Seifert, 2019).

Taken together, these works suggest that semi-quenched entropy is not a single standardized quantity but a recurrent structural motif. In large deviations it is an explicit interpolation between environment reshaping and dynamic tilting. In disordered entanglement it is an operational regime in which quenched randomness is encoded in deterministic effective propagators before entropy is computed. In semi-local holography it describes a state with quenched thermal entropy density but nontrivial entanglement geometry. In thermally isolated dynamics it names the regime in which quenches are weak enough that subextensive entropy corrections remain visible. The common feature is the same: some sector is frozen, typical, or only softly penalized, while another sector continues to carry the entropy-producing dynamics.