--- title: Temperature-Dependent Gaussian Packets url: https://www.emergentmind.com/topics/temperature-dependent-gaussian-packets-gpps type: topic --- # Temperature-Dependent Gaussian Packets 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 [1609.00014] [1104.1256] [1210.2828] [2507.13960]. 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 \[ \lambda = \sqrt{\frac{2\pi\hbar^{2}\beta}{m}} = \sqrt{\frac{2\pi\hbar^{2}}{m k_{\rm B}T}}. \] The corresponding normalized static packet is \[ \phi_{\mathbf r,\lambda}(\mathbf r')=\frac{2\sqrt{2}}{\lambda^{3/2}}\exp\!\left[-2\pi\frac{|\mathbf r-\mathbf r'|^2}{\lambda^2}\right], \] and the one-particle thermal state admits the exact convex decomposition \[ \rho_{\mathrm{th}}^{(1)}=\int_V \frac{d\mathbf r}{V}\, \ket{\Phi_{\mathbf r,\lambda}}\bra{\Phi_{\mathbf r,\lambda}}. \] Each such packet has zero mean momentum, spatial width \[ \Delta x=\Delta y=\Delta z=\frac{\lambda}{\sqrt{8\pi}}, \] momentum width \[ \Delta p_x=\Delta p_y=\Delta p_z=\frac{\sqrt{2\pi}\hbar}{\lambda}, \] and kinetic energy contribution \[ \frac{\Delta p_x^2+\Delta p_y^2+\Delta p_z^2}{2m}=\frac{3}{2}k_{\rm B}T. \] A more general decomposition introduces nonzero mean momentum. The packet \[ \phi_{\mathbf r\mathbf p,\lambda_s}(\mathbf r') =\frac{2\sqrt{2}}{\lambda_s^{3/2}} \exp\!\left[\frac{i}{\hbar}\mathbf p\cdot(\mathbf r'-\mathbf r)\right] \exp\!\left[-2\pi\frac{|\mathbf r'-\mathbf r|^2}{\lambda_s^2}\right] \] is weighted by the Gaussian momentum distribution \[ f_{\lambda_m}(\mathbf p)=\left(\frac{\lambda_m}{2\pi\hbar}\right)^3 \exp\!\left[-\frac{\lambda_m^2 p^2}{4\pi\hbar^2}\right], \] with \[ \lambda^{-2}=\lambda_s^{-2}+\lambda_m^{-2}, \qquad T=T_s+T_m. \] The one-particle thermal state becomes \[ \rho_{\mathrm{th}}^{(1)} =\int_V\frac{d\mathbf r}{V}\int d\mathbf p\; f_{\lambda_m}(\mathbf p)\, \ket{\Phi_{\mathbf r\mathbf p,\lambda_s}} \bra{\Phi_{\mathbf r\mathbf p,\lambda_s}}. \] Here \(T_s\) governs the packet width and \(T_m\) the momentum distribution. The thermal energy is correspondingly split between the packet’s internal width and its mean momentum [1609.00014]. The \(N\)-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 \(\frac{1}{N!}(V/\lambda^3)^N\), and the ensemble becomes effectively classical [1609.00014]. ## 3. Gaussian partition functions and imaginary-time propagation In semiclassical statistical mechanics, GPPs appear as Gaussian approximations to the Boltzmann operator \[ K(\beta)=e^{-\beta \hat H},\qquad Z(\beta)=\mathrm{Tr}\,e^{-\beta \hat H}. \] The central construction is the frozen Gaussian imaginary-time propagator, where the Gaussian width matrix \(\mathbf\Gamma\) is held fixed and only the Gaussian centers \(\mathbf q(\tau)\) evolve according to \[ \frac{\partial \mathbf q(\tau)}{\partial \tau} = -\mathbf\Gamma^{-1}\langle \nabla V(\mathbf q(\tau))\rangle. \] The resulting zeroth-order approximation \(K_0(\beta)\) yields a Gaussian partition function \(Z_0(\beta)=\mathrm{Tr}[K_0(\beta)]\). Temperature dependence enters explicitly through \(\beta\) and implicitly through the imaginary-time trajectories propagated up to \(\tau=\beta/2\) or \(\beta\) [1104.1256]. The frozen Gaussian propagator is the zeroth term of a correction series. The correction operator is \[ C(\tau)=-\frac{\partial}{\partial \tau}K_0(\tau)-\hat H K_0(\tau), \] and the first-order corrected partition function is \[ Z_1(\beta)=Z_0(\beta)+Z_{\mathrm C1}(\beta). \] A central diagnostic is the relative correction \[ R(\beta;\mathbf\Gamma)=\frac{|Z_{\mathrm C1}(\beta;\mathbf\Gamma)|}{Z_0(\beta;\mathbf\Gamma)}. \] When this ratio is small, the Gaussian approximation is reliable; when it becomes order \(1\) or larger, the approximation ceases to be trustworthy [1104.1256]. For the argon trimer, the paper uses a structured \(9\times 9\) width matrix with parameters \(D_1\) and \(D_2\), a Gaussian fit to the Morse interaction, and confinement \(R_c=10\,\mathrm{\AA}\). In the bound regime \(T\lesssim 30\,\mathrm{K}\), the ratio \(Z_{\mathrm C1}/Z_0\) is minimized for \(D_2\) in the range \(20\)–\(30\,\mathrm{\AA^{-2}}\), with the optimal around \(D_2=25\,\mathrm{\AA^{-2}}\). For \(R_c=10\,\mathrm{\AA}\), the minimum with respect to \(D_1\) occurs around \(D_1\approx 0.5\)–\(1\,\mathrm{\AA^{-2}}\). The first-order correction improves the frozen Gaussian energy and specific heat in the transition region and above, while the ratio \(Z_{\mathrm C1}/Z_0\) signals breakdown near \(T\approx 12\,\mathrm{K}\). The physically important bound-to-dissociated transition occurs in the range about \(25\)–\(35\) K, where the corrected Gaussian treatment remains accurate [1104.1256]. This framework treats GPPs as temperature-dependent Gaussian representations of \(e^{-\beta \hat H}\). 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 \(n=2m+1\) monochromatic modes with \[ \omega_{j,k}=\bar\omega_j+k\delta,\qquad k=-m,\dots,m,\qquad \Delta\omega_j=2m\delta. \] For realistic measurements that do not resolve single microscopic modes, the relevant observables are collective annihilation operators, such as \[ \hat A_{\mathbf 1}(0)=\frac{1}{\sqrt{2m+1}}\sum_{k=-m}^{m}\hat a_{1,k}, \qquad \hat A_{\mathbf 2}^\dagger(0)=\frac{1}{\sqrt{2m+1}}\sum_{k=-m}^{m}\hat a_{2,k}^\dagger. \] 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 \(4\times 4\) covariance matrix \(\Sigma(t)\) [1210.2828]. Finite temperature is introduced by taking every microscopic mode initially in a thermal state with Bose–Einstein occupation \[ \bar n_{j0,k}=\frac{1}{e^{\beta_{j,k}}-1}, \qquad \beta_{j,k}=\frac{\hbar \omega_{j,k}}{k_B T}. \] The macro-modes are then temperature-dependent Gaussian packets whose initial variances are determined by the collective thermal photon numbers \(\bar N_{10}(T)\) and \(\bar N_{20}(T)\). Entanglement is quantified by the logarithmic negativity \[ \mathcal E_N=\max\{0,-\ln \tilde\nu\}, \] and the separability criterion can be written as \[ \mathcal S(t)=\mathcal S_0-|\det\gamma(t)|<0, \] with thermal noise floor \[ \mathcal S_0= \bar N_{10}\bar N_{20}(\bar N_{10}+1)(\bar N_{20}+1). \] Entanglement therefore appears only when the generated cross-correlations exceed a temperature-dependent threshold [1210.2828]. 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 \(n\) in the pairwise case and coincides with the \(n=1\) case, whereas it depends explicitly on \(n\) 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 \(n\). At \(T=30^\circ\,\mathrm{K}\) and \(\tau_i=wt=0.3324\), the reported fits are \[ \mathcal E_N=0.963\,n \quad \text{(vacuum)}, \qquad \mathcal E_N=0.963\,(n-5)\quad \text{(thermal)}. \] The birth time of entanglement is finite for thermal initial states and increases with temperature. In the pairwise case it is independent of \(n\); in the one-to-all case it decreases with \(n\). For fixed interaction time, the critical temperature for observing entanglement is approximately \(4.38\,\mathrm{K}\) in the one-to-all case with \(n=1\), and approximately \(1200\,\mathrm{K}\) for \(n=5\). The graph-theoretic interpretation is explicit: the one-to-all interaction graph is connected with vertex degree \(n\), whereas the pairwise graph decomposes into \(n\) disconnected two-vertex components with degree \(1\). In this model, stronger connectivity yields stronger and more thermally robust entanglement of the collective Gaussian packets [1210.2828]. ## 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 \[ f(z,t)=\frac{1}{Z(t)}\exp\!\left[ -\frac{1}{2}(z-\bar z(t))^\top \boldsymbol\Sigma^{-1}(t)(z-\bar z(t)) \right], \] and in practice it is approximated by a product of single-atom Gaussians \[ f(z,t)=\frac{1}{M}\prod_{i=1}^N f_i(z_i,t). \] Each atom \(i\) is characterized by mean position \(\bar q_i\), mean momentum \(\bar p_i\), position covariance \(\boldsymbol\Sigma_i^{(q,q)}\), momentum covariance \(\boldsymbol\Sigma_i^{(p,p)}\), and position–momentum covariance \(\boldsymbol\Sigma_i^{(p,q)}\) [2507.13960]. For 2D amorphous silica, the anisotropic formulation uses \[ \boldsymbol\Sigma_i^{(q,q)} = \begin{bmatrix} \Sigma_i^{xx} & \Sigma_i^{xy}\\ \Sigma_i^{xy} & \Sigma_i^{yy} \end{bmatrix}, \qquad \Omega_i^{\alpha\beta}=m_i k_B T\,\delta_{\alpha\beta}, \] and the quasistatic equilibrium equations become \[ \langle f_i\rangle=\mathbf 0, \] together with the covariance balance equations \[ \frac{\Omega^{\alpha \beta}_i}{m_i}\delta_{\alpha\beta} + \frac{ \langle f^\alpha_i(q)(q^\beta_i-\bar q^\beta_i)\rangle + \langle f^\beta_i(q)(q^\alpha_i-\bar q^\alpha_i)\rangle }{2} =0. \] 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 [2507.13960]. The numerical loading protocol is the finite-temperature analog of AQS. At a given \(T\), a zero-stress reference state \(\{\bar q^0(T),\boldsymbol\Sigma^{(q,q),0}(T),\boldsymbol H^0(T)\}\) is obtained with FIRE. A small affine strain increment is applied, \[ \Delta\varepsilon= \begin{bmatrix} 0 & 0\\ 0 & \gamma/L_y \end{bmatrix}, \] the cell is updated, and the GPP equations are re-solved using third-order Gaussian quadrature aligned with the principal axes of \(\Sigma_i^{(q,q)}\). 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 [2507.13960]. The positional covariance is also used as a predictor of local instability. The scalar measure \[ \Sigma_i^2=\det \boldsymbol\Sigma_i^{(q,q)} \] identifies atoms with large directional thermal fluctuations. In the undeformed thermally expanded state of 2D silica, the two oxygen atoms with the largest \(\Sigma_i^2\) 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 \(\Sigma_i^2\) in the preceding snapshot. The anisotropic formulation is therefore essential for localizing failure-prone zones accurately [2507.13960]. The method also reproduces thermal expansion. Over \(T^*\in[0,10^{-3}]\), both isotropic and anisotropic GPP reproduce molecular-dynamics box dimensions and mean potential energy accurately; at the highest \(T\), errors in \(L_x\) and average energy are less than \(0.1\%\) in the isotropic formulation and are reduced by about a factor of \(2\) in the anisotropic formulation, while errors in \(L_y\) are about \(0.2\%\) 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 \[ A(x|y)=\min\left[1,\exp\!\left(-\frac{U(x)-U(y)}{k_B T}\right)\right]. \] With \(N_s=N_p N\) trial moves, the last accepted state seeds the next GPP relaxation. Acceptance rates are reported as \(60\)–\(70\%\), and the combinations \(\{\gamma,N_p\}=\{10^{-4},5\cdot10^3\}\) and \(\{10^{-4},10^4\}\) reproduce onset stresses comparable to low-strain-rate molecular dynamics [2507.13960]. ## 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 \[ g_z^{C,\varepsilon}(x) = (\pi\varepsilon)^{-d/4}\det(\operatorname{Im}C)^{1/4} \exp\!\left[ \frac{i}{\varepsilon}\left( \frac12(x-q)^T C(x-q)+p^T(x-q) \right) \right], \] with exact inversion \[ \psi = (2\pi\varepsilon)^{-d} \int_{\mathbb R^{2d}} \langle g_z\mid \psi\rangle\, g_z\,dz. \] A Gaussian partition of unity is built from \[ S(x)=\sum_{k\in\Gamma_q}|g_0(x-q_k)|^2, \] 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 \[ E^{(\mathrm{GH})}