Temperature-Dependent Gaussian Packets

• Temperature-dependent Gaussian packets are Gaussian representations whose parameters, such as width and covariance, incorporate temperature effects to model thermal systems.
• They are applied in diverse settings including thermal density operator decompositions, semiclassical Boltzmann approximations, and finite-temperature entanglement analyses in quantum and material physics.
• The methods provide actionable insights through error analysis, optimal parameter tuning, and correction protocols that improve the accuracy of simulations from quantum gases to disordered solids.

Temperature-dependent Gaussian packets (GPPs) are Gaussian states, distributions, or ansätze whose widths, covariance matrices, or statistical weights depend explicitly on temperature. In the literature, this designation spans several technically distinct constructions: convex decompositions of thermal density operators into Gaussian wave packets, semiclassical Gaussian approximations to the Boltzmann operator and partition function, collective Gaussian macro-modes in finite-temperature multimode parametric down-conversion, and quasistatic Gaussian phase packets for finite-temperature mechanics of disordered solids (Chenu et al., 2016, Cartarius et al., 2011, Martins, 2012, Spínola et al., 18 Jul 2025). Across these settings, temperature enters through thermal de Broglie wavelengths, Bose–Einstein occupations, inverse-temperature imaginary-time propagation, or free-energy-minimizing phase-space covariances.

1. Terminological scope and shared structure

The literature does not use a single standardized formalism for GPPs. Instead, it develops Gaussian packet descriptions in different physical and mathematical settings, each of which makes temperature dependence explicit in a different variable set.

Setting	Gaussian object	Temperature dependence
Thermal equilibrium of non-interacting particles	Wave packets $\phi_{\mathbf r,\lambda}$ and $\phi_{\mathbf r\mathbf p,\lambda_s}$	Thermal de Broglie wavelengths $\lambda,\lambda_s,\lambda_m$
Semiclassical statistical mechanics	Frozen or thawed Gaussian imaginary-time propagators	Inverse temperature $\beta$ in $K(\beta)=e^{-\beta \hat H}$
Finite-temperature MPDC	Collective macro-modes of signal and idler wave-packets	Thermal occupations $\bar n_{j0,k}(T)$, macro-occupations $\bar N_{10},\bar N_{20}$
Quasistatic mechanics of disordered solids	Single-atom Gaussian phase packets $f_i(z_i,t)$	Momentum variance $m_i k_B T$ and temperature-dependent $\Sigma_i^{(q,q)}$

A common structural feature is that the state is represented by Gaussian data in either configuration space, momentum space, phase space, or collective mode space. This suggests that GPPs are best understood as a class of temperature-aware Gaussian representations rather than as a single named model.

2. Thermal equilibrium as an ensemble of Gaussian wave packets

For non-interacting, non-relativistic bosons or fermions in a large box, the canonical thermal density operator can be decomposed into localized Gaussian wave packets instead of delocalized energy eigenstates. The thermal de Broglie wavelength is

$\phi_{\mathbf r\mathbf p,\lambda_s}$0

The corresponding normalized static packet is

$\phi_{\mathbf r\mathbf p,\lambda_s}$1

and the one-particle thermal state admits the exact convex decomposition

$\phi_{\mathbf r\mathbf p,\lambda_s}$2

Each such packet has zero mean momentum, spatial width

$\phi_{\mathbf r\mathbf p,\lambda_s}$3

momentum width

$\phi_{\mathbf r\mathbf p,\lambda_s}$4

and kinetic energy contribution

$\phi_{\mathbf r\mathbf p,\lambda_s}$5

A more general decomposition introduces nonzero mean momentum. The packet

$\phi_{\mathbf r\mathbf p,\lambda_s}$6

is weighted by the Gaussian momentum distribution

$\phi_{\mathbf r\mathbf p,\lambda_s}$7

with

$\phi_{\mathbf r\mathbf p,\lambda_s}$8

The one-particle thermal state becomes

$\phi_{\mathbf r\mathbf p,\lambda_s}$9

Here $\lambda,\lambda_s,\lambda_m$0 governs the packet width and $\lambda,\lambda_s,\lambda_m$1 the momentum distribution. The thermal energy is correspondingly split between the packet’s internal width and its mean momentum (Chenu et al., 2016).

The $\lambda,\lambda_s,\lambda_m$2-particle generalization uses products of such packets with the correct bosonic or fermionic field-operator ordering, so the resulting wave functions are automatically symmetrized or antisymmetrized. Their norms encode quantum statistics and yield the standard first quantum corrections to the partition function. In the Maxwell–Boltzmann limit, when the packet overlap is negligible, the norm corrections vanish, the partition function reduces to $\lambda,\lambda_s,\lambda_m$3, and the ensemble becomes effectively classical (Chenu et al., 2016).

3. Gaussian partition functions and imaginary-time propagation

In semiclassical statistical mechanics, GPPs appear as Gaussian approximations to the Boltzmann operator

$\lambda,\lambda_s,\lambda_m$4

The central construction is the frozen Gaussian imaginary-time propagator, where the Gaussian width matrix $\lambda,\lambda_s,\lambda_m$5 is held fixed and only the Gaussian centers $\lambda,\lambda_s,\lambda_m$6 evolve according to

$\lambda,\lambda_s,\lambda_m$7

The resulting zeroth-order approximation $\lambda,\lambda_s,\lambda_m$8 yields a Gaussian partition function $\lambda,\lambda_s,\lambda_m$9. Temperature dependence enters explicitly through $\beta$0 and implicitly through the imaginary-time trajectories propagated up to $\beta$1 or $\beta$2 (Cartarius et al., 2011).

The frozen Gaussian propagator is the zeroth term of a correction series. The correction operator is

$\beta$3

and the first-order corrected partition function is

$\beta$4

A central diagnostic is the relative correction

$\beta$5

When this ratio is small, the Gaussian approximation is reliable; when it becomes order $\beta$6 or larger, the approximation ceases to be trustworthy (Cartarius et al., 2011).

For the argon trimer, the paper uses a structured $\beta$7 width matrix with parameters $\beta$8 and $\beta$9, a Gaussian fit to the Morse interaction, and confinement $K(\beta)=e^{-\beta \hat H}$0. In the bound regime $K(\beta)=e^{-\beta \hat H}$1, the ratio $K(\beta)=e^{-\beta \hat H}$2 is minimized for $K(\beta)=e^{-\beta \hat H}$3 in the range $K(\beta)=e^{-\beta \hat H}$4–$K(\beta)=e^{-\beta \hat H}$5, with the optimal around $K(\beta)=e^{-\beta \hat H}$6. For $K(\beta)=e^{-\beta \hat H}$7, the minimum with respect to $K(\beta)=e^{-\beta \hat H}$8 occurs around $K(\beta)=e^{-\beta \hat H}$9–$\bar n_{j0,k}(T)$0. The first-order correction improves the frozen Gaussian energy and specific heat in the transition region and above, while the ratio $\bar n_{j0,k}(T)$1 signals breakdown near $\bar n_{j0,k}(T)$2. The physically important bound-to-dissociated transition occurs in the range about $\bar n_{j0,k}(T)$3–$\bar n_{j0,k}(T)$4 K, where the corrected Gaussian treatment remains accurate (Cartarius et al., 2011).

This framework treats GPPs as temperature-dependent Gaussian representations of $\bar n_{j0,k}(T)$5. The temperature dependence is not only part of the model input but also part of the internal error analysis.

4. Collective Gaussian packets and macroscopic entanglement

In finite-bandwidth multimode parametric down-conversion, GPPs arise as collective Gaussian wave-packets built from many unresolved microscopic modes. The signal and idler packets each contain $\bar n_{j0,k}(T)$6 monochromatic modes with

$\bar n_{j0,k}(T)$7

For realistic measurements that do not resolve single microscopic modes, the relevant observables are collective annihilation operators, such as

$\bar n_{j0,k}(T)$8

Because the Hamiltonians are bilinear and the initial states are vacuum or thermal, the collective macro-modes remain in Gaussian states at all times and are completely characterized by a $\bar n_{j0,k}(T)$9 covariance matrix $\bar N_{10},\bar N_{20}$0 (Martins, 2012).

Finite temperature is introduced by taking every microscopic mode initially in a thermal state with Bose–Einstein occupation

$\bar N_{10},\bar N_{20}$1

The macro-modes are then temperature-dependent Gaussian packets whose initial variances are determined by the collective thermal photon numbers $\bar N_{10},\bar N_{20}$2 and $\bar N_{10},\bar N_{20}$3. Entanglement is quantified by the logarithmic negativity

$\bar N_{10},\bar N_{20}$4

and the separability criterion can be written as

$\bar N_{10},\bar N_{20}$5

with thermal noise floor

$\bar N_{10},\bar N_{20}$6

Entanglement therefore appears only when the generated cross-correlations exceed a temperature-dependent threshold (Martins, 2012).

Two interaction topologies are contrasted. In the pairwise pattern, each signal mode couples only to its energy-matched idler partner; in the one-to-all pattern, each signal mode couples to all idler modes. The macroscopic covariance matrix is independent of $\bar N_{10},\bar N_{20}$7 in the pairwise case and coincides with the $\bar N_{10},\bar N_{20}$8 case, whereas it depends explicitly on $\bar N_{10},\bar N_{20}$9 in the one-to-all case. Consequently, in the pairwise interaction the degree of macroscopic entanglement is independent of the number of modes, while in the one-to-all interaction it increases linearly with $f_i(z_i,t)$0. At $f_i(z_i,t)$1 and $f_i(z_i,t)$2, the reported fits are

$f_i(z_i,t)$3

The birth time of entanglement is finite for thermal initial states and increases with temperature. In the pairwise case it is independent of $f_i(z_i,t)$4; in the one-to-all case it decreases with $f_i(z_i,t)$5. For fixed interaction time, the critical temperature for observing entanglement is approximately $f_i(z_i,t)$6 in the one-to-all case with $f_i(z_i,t)$7, and approximately $f_i(z_i,t)$8 for $f_i(z_i,t)$9. The graph-theoretic interpretation is explicit: the one-to-all interaction graph is connected with vertex degree $m_i k_B T$0, whereas the pairwise graph decomposes into $m_i k_B T$1 disconnected two-vertex components with degree $m_i k_B T$2. In this model, stronger connectivity yields stronger and more thermally robust entanglement of the collective Gaussian packets (Martins, 2012).

5. Gaussian phase packets in finite-temperature quasistatic mechanics

A distinct meaning of GPP is introduced in finite-temperature quasistatic mechanics of disordered solids, where each atom is represented by a Gaussian probability density in phase space rather than by a point microstate. The full ansatz is

$m_i k_B T$3

and in practice it is approximated by a product of single-atom Gaussians

$m_i k_B T$4

Each atom $m_i k_B T$5 is characterized by mean position $m_i k_B T$6, mean momentum $m_i k_B T$7, position covariance $m_i k_B T$8, momentum covariance $m_i k_B T$9, and position–momentum covariance $\Sigma_i^{(q,q)}$0 (Spínola et al., 18 Jul 2025).

For 2D amorphous silica, the anisotropic formulation uses

$\Sigma_i^{(q,q)}$1

and the quasistatic equilibrium equations become

$\Sigma_i^{(q,q)}$2

together with the covariance balance equations

$\Sigma_i^{(q,q)}$3

These equations are equivalent to stationarity of the free energy under the Gaussian ansatz. Temperature enters explicitly through the canonical momentum variance and implicitly through the position covariance that is determined self-consistently by local curvature of the potential energy landscape (Spínola et al., 18 Jul 2025).

The numerical loading protocol is the finite-temperature analog of AQS. At a given $\Sigma_i^{(q,q)}$4, a zero-stress reference state $\Sigma_i^{(q,q)}$5 is obtained with FIRE. A small affine strain increment is applied,

$\Sigma_i^{(q,q)}$6

the cell is updated, and the GPP equations are re-solved using third-order Gaussian quadrature aligned with the principal axes of $\Sigma_i^{(q,q)}$7. Positive definiteness is enforced by monitoring eigenvalues and reducing the FIRE time step if necessary. Macroscopic stress is then computed from a virial-like expression (Spínola et al., 18 Jul 2025).

The positional covariance is also used as a predictor of local instability. The scalar measure

$\Sigma_i^{(q,q)}$8

identifies atoms with large directional thermal fluctuations. In the undeformed thermally expanded state of 2D silica, the two oxygen atoms with the largest $\Sigma_i^{(q,q)}$9 coincide with the two sites where molecular dynamics shows the first Si–O bond breaking. During loading, snapshots taken just before each bond-breaking event show that the next broken bond always involves an atom with the first- or second-highest $\phi_{\mathbf r\mathbf p,\lambda_s}$00 in the preceding snapshot. The anisotropic formulation is therefore essential for localizing failure-prone zones accurately (Spínola et al., 18 Jul 2025).

The method also reproduces thermal expansion. Over $\phi_{\mathbf r\mathbf p,\lambda_s}$01, both isotropic and anisotropic GPP reproduce molecular-dynamics box dimensions and mean potential energy accurately; at the highest $\phi_{\mathbf r\mathbf p,\lambda_s}$02, errors in $\phi_{\mathbf r\mathbf p,\lambda_s}$03 and average energy are less than $\phi_{\mathbf r\mathbf p,\lambda_s}$04 in the isotropic formulation and are reduced by about a factor of $\phi_{\mathbf r\mathbf p,\lambda_s}$05 in the anisotropic formulation, while errors in $\phi_{\mathbf r\mathbf p,\lambda_s}$06 are about $\phi_{\mathbf r\mathbf p,\lambda_s}$07 isotropic and about half that anisotropic. To incorporate thermally activated basin changes, the paper combines GPP with a Metropolis sampler. Proposed moves are drawn from the configuration-space GPP density, and acceptance is

$\phi_{\mathbf r\mathbf p,\lambda_s}$08

With $\phi_{\mathbf r\mathbf p,\lambda_s}$09 trial moves, the last accepted state seeds the next GPP relaxation. Acceptance rates are reported as $\phi_{\mathbf r\mathbf p,\lambda_s}$10–$\phi_{\mathbf r\mathbf p,\lambda_s}$11, and the combinations $\phi_{\mathbf r\mathbf p,\lambda_s}$12 and $\phi_{\mathbf r\mathbf p,\lambda_s}$13 reproduce onset stresses comparable to low-strain-rate molecular dynamics (Spínola et al., 18 Jul 2025).

6. Mathematical packet representations and fluctuation limits

Gaussian packet methods also possess a mathematical infrastructure that is compatible with temperature dependence, even when temperature is not the original organizing variable. The Gaussian wave packet transform uses

$\phi_{\mathbf r\mathbf p,\lambda_s}$14

with exact inversion

$\phi_{\mathbf r\mathbf p,\lambda_s}$15

A Gaussian partition of unity is built from

$\phi_{\mathbf r\mathbf p,\lambda_s}$16

and the momentum integral can be discretized by infinite Riemann sums, truncated midpoint rules, or Gauss–Hermite quadrature. The Gauss–Hermite representation reduces the number of basis functions significantly and achieves error bounds of the form

$\phi_{\mathbf r\mathbf p,\lambda_s}$17

A plausible implication is that if $\phi_{\mathbf r\mathbf p,\lambda_s}$18 or $\phi_{\mathbf r\mathbf p,\lambda_s}$19 are made temperature dependent, then the same partition-of-unity and quadrature machinery yields temperature-adapted Gaussian packet bases with temperature-dependent grid scales in position and momentum (Bergold et al., 2020).

A different large-scale Gaussian phenomenon appears in Coulomb gases. There, localized linear statistics become Gaussian above the temperature-dependent minimal rigidity scale $\phi_{\mathbf r\mathbf p,\lambda_s}$20, with $\phi_{\mathbf r\mathbf p,\lambda_s}$21 determined by the effective inverse temperature. In dimension $\phi_{\mathbf r\mathbf p,\lambda_s}$22, the resulting central limit theorem identifies a Gaussian free field covariance for the fluctuation potential, while in higher dimensions analogous Gaussian behavior is obtained conditionally on regularity of the free-energy expansion. This suggests a fluctuation-theoretic notion of temperature-dependent Gaussian packets: localized density modes behave as Gaussian objects whose admissible spatial support and fluctuation amplitude are both controlled by temperature (Serfaty, 2020).

Taken together, these mathematical results show that temperature dependence can affect GPPs at three levels: the packet width, the quadrature or basis resolution needed to represent them, and the spatial scale above which Gaussian fluctuation laws apply.