From 9365065ce556e7f48e9014b82c4c22e7d351dc68 Mon Sep 17 00:00:00 2001
From: Anton Akhmerov <anton.akhmerov@gmail.com>
Date: Mon, 24 Mar 2025 11:29:55 +0100
Subject: [PATCH] expand 2.4 solution

---
 docs/2_debye_model_solutions.md |   98 +-
 pixi.lock                       | 5621 ++++++++-----------------------
 2 files changed, 1510 insertions(+), 4209 deletions(-)

diff --git a/docs/2_debye_model_solutions.md b/docs/2_debye_model_solutions.md
index 0d9bbff..5dfe740 100644
--- a/docs/2_debye_model_solutions.md
+++ b/docs/2_debye_model_solutions.md
@@ -32,14 +32,14 @@ configure_plotting()
 ### Warm-up exercises
 
 1. For low T, $1/T \rightarrow \infty$. The heat capacity is then given as:
-    
+
     $$
     C \overset{\mathrm{low \: T}}{\approx} 9Nk_{\mathrm{B}}\left(\frac{T}{T_{D}}\right)^3\int_0^{\infty}\frac{x^4{\mathrm{e}}^x}{({\mathrm{e}}^x-1)^2}{\mathrm{d}}x.
     $$
 
 2. See plot below (shown for $T_{D,1} < T_{D,2}$)
 3. The polarization describes the direction of the motion of the atoms in the wave with respect to the direction in which the wave travels. In 3D, there are only 3 different polarizations possible.
-4. The integral can be expressed as     
+4. The integral can be expressed as
     $$
     \int dk_x dk_y = \int k dk d\phi
     $$
@@ -68,18 +68,19 @@ ax.set_yticklabels(['$0$'])
 ax.legend();
 ```
 
-### Exercise 1: Deriving the density of states for the linear dispersion relation of the Debye model.
+### Exercise 1: Deriving the density of states for the linear dispersion relation of the Debye model
+
 1. $\omega = v_s|\mathbf{k}|$
 2. The distance between nearest-neighbour points in $\mathbf{k}$-space is $2\pi/L$. The density of $\mathbf{k}$-points in 1, 2, and 3 dimensions is $L/(2\pi)$, $L^2/(2\pi)^2$, and $L^3/(2\pi)^3$ respectively.
 3. Express the number of states between frequencies $0<\omega<\omega_0$ as an integral over k-space. Do so for 1D, 2D and 3D. Do not forget the possible polarizations. We assume that in $d$ dimensions there are $d_p$ polarizations.
-    
+
     \begin{align*}
     N_\text{states, 1D} & = 1_p \frac{L}{2\pi} \int_0^{k_0} 2 dk \\
     N_\text{states, 2D} & = 2_p \left(\frac{L}{2\pi}\right)^2 \int_0^{k_0} 2\pi k dk \\
-    N_\text{states, 3D} & = 3_p  \left(\frac{L}{2\pi}\right)^3 \int_0^{k_0} 4\pi k^2 dk 
+    N_\text{states, 3D} & = 3_p  \left(\frac{L}{2\pi}\right)^3 \int_0^{k_0} 4\pi k^2 dk
     \end{align*}
 
-4.   We use $k=\mathbf{k}| = \omega/v_s$ and $dk = d\omega/v_s$ to get 
+4. We use $k=\mathbf{k}| = \omega/v_s$ and $dk = d\omega/v_s$ to get
 
     \begin{align*}
     N_\text{states, 1D} & = 1_p \frac{L}{2\pi} \int_0^{\omega_0} 2 \frac{1}{v_s} d\omega  := \int_0^{\omega_0} g_{1D}(\omega) d\omega \\
@@ -91,52 +92,52 @@ ax.legend();
 
 5. The density of states is the number of states per unit frequency. It has units of 1 over frequency
 
-###  Exercise 2: Debye model in 2D.
+### Exercise 2: Debye model in 2D
+
 1. The energy stored in the vibrational modes of a two-dimensional Debye solid is:
-    
+
     \begin{align*}
-    E & = 2_p \frac{L^2}{4\pi^2}\int_0^{\infty}(n_B(\omega(\mathbf{k}))+\frac{1}{2})\hbar\omega(\mathbf{k}) d\mathbf{k} \\    
+    E & = 2_p \frac{L^2}{4\pi^2}\int_0^{\infty}(n_B(\omega(\mathbf{k}))+\frac{1}{2})\hbar\omega(\mathbf{k}) d\mathbf{k} \\
     & = \frac{L^2}{\pi v^2\hbar^2\beta^3}\int_{0}^{\beta\hbar\omega_D}\frac{x^2}{e^{x} - 1}dx + E_0
     \end{align*}
 
-  
 2. The high-$T$ limit implies $\beta \rightarrow 0$. Therefore, $n_B \approx k_B T/\hbar\omega$, and the integral becomes particularly illuminating:
-    
+
     $$
     E = \int_{0}^{\omega_D} \hbar\omega  n_B(\omega) g(\omega) d\omega \approx \int_{0}^{\omega_D} k_B T g(\omega) d\omega = N_\text{modes} k_B T
     $$
 
     where we neglected the zero-point energy. In 2D, we have $N_\text{modes} = 2_p N_\text{atoms}$, so that we recover the 2D law of Dulong–Petit $C_v = dE/dT = 2 k_B$ per atom.
 
-3.  In the low temperature limit, the high-energy modes are not excited so we can safely let the upper boundary of the integral go to infinity. For convenience, we write $g(\omega) = \alpha \omega$, with $\alpha = \frac{L^2}{\pi v_s^2}$. We get 
-    
-    $$ 
-    E = \int_0^{\omega_D}\frac{\hbar\omega g(\omega)}{e^{\hbar\omega/k_B T}- 1}d\omega \approx \alpha\frac{k^3 T^3}{\hbar^2}\int_0^\infty\frac{x^2}{e^x-1}dx 
+3. In the low temperature limit, the high-energy modes are not excited so we can safely let the upper boundary of the integral go to infinity. For convenience, we write $g(\omega) = \alpha \omega$, with $\alpha = \frac{L^2}{\pi v_s^2}$. We get
+
     $$
-    
-    From which we find  $C_v = dE/dT = K T^2$, with 
-    
+    E = \int_0^{\omega_D}\frac{\hbar\omega g(\omega)}{e^{\hbar\omega/k_B T}- 1}d\omega \approx \alpha\frac{k^3 T^3}{\hbar^2}\int_0^\infty\frac{x^2}{e^x-1}dx
     $$
-    K = 3\alpha\frac{k^3}{\hbar^2}\int_0^\infty\frac{x^2}{e^x-1}dx 
+
+    From which we find  $C_v = dE/dT = K T^2$, with
+
     $$
+    K = 3\alpha\frac{k^3}{\hbar^2}\int_0^\infty\frac{x^2}{e^x-1}dx
+    $$
+
+### Exercise 3: Longitudinal and transverse vibrations with different sound velocities
+
+1. The key idea is that the total energy in the individual harmonic oscillators (the vibrational modes) is the sum of the energies in the individual oscillators: $E = \int_0^{\omega_D}\frac{\hbar \omega g(\omega)}{e^{\beta\hbar\omega} - 1}d\omega + E_Z$, where $g(\omega) = g_\parallel(\omega) + g_\perp(\omega)$. Using
 
-###  Exercise 3: Longitudinal and transverse vibrations with different sound velocities
-1. The key idea is that the total energy in the individual harmonic oscillators (the vibrational modes) is the sum of the energies in the individual oscillators: $E = \int_0^{\omega_D}\frac{\hbar \omega g(\omega)}{e^{\beta\hbar\omega} - 1}d\omega + E_Z$, where $g(\omega) = g_\parallel(\omega) + g_\perp(\omega)$. Using 
-    
     \begin{align*}
     g_\parallel & = 1_p  \frac{L^3}{2\pi^2} \frac{\omega^2}{v_\parallel^3}, \\
     g_\perp     & = 2_p  \frac{L^3}{2\pi^2} \frac{\omega^2}{v_\perp^3},
     \end{align*}
 
-    we get 
-    
-    $$ 
+    we get
+
+    $$
     E = \frac{L^3 k_B^4 T^4}{2\pi^2\hbar^3}\left(\frac{2}{v_\perp^3} + \frac{1}   {v_\parallel^3}\right)\int_{0}^{\beta\hbar\omega_D}\frac{x^3}{e^{x} - 1}dx.
     $$
 
+2. As in exercise 1, in the high-T limit, we have $\beta \rightarrow 0. Therefore, $n_B \approx k_B T/\hbar\omega$ and the integral for $E$ becomes:
 
-2. As in exercise 1, in the high-T limit, we have $\beta \rightarrow 0$. Therefore, $n_B \approx k_B T/\hbar\omega$ and the integral for $E$ becomes: 
-    
     $$
     E = \int_{0}^{\omega_D} \hbar\omega  n_B(\omega) g(\omega) d\omega \approx \int_{0}^{\omega_D} k_B T g(\omega) d\omega +E_Z = N_\text{modes} k_B T + E_Z
     $$
@@ -144,18 +145,17 @@ ax.legend();
     and we are left with the Dulong-Petit law $C = 3N_\text{atoms} k_B$.
 
 3. In the low temperature limit, we can let the upper integral boundary go to infinity as in exercise 1. This yields
-    
+
     $$
     C \approx \frac{2\pi^2 k_B^4 L^3}{15\hbar^3}\left(\frac{2}{v_\perp^3} + \frac{1}{v_\parallel^3}\right)T^3
     $$
-    
+
     where we used $\int_{0}^{\infty}\frac{x^3}{e^{x} - 1}dx = \frac{\pi^4}{15}$.
-    
 
+### Exercise 4: Anisotropic sound velocities
 
-### Exercise 4: Anisotropic sound velocities.
 In this case, the velocity depends on the direction. Note however that, in contrast with the previous exercise, the polarization does not affect the dispersion of the waves. We get
-    
+
 \begin{align*}
 E =& 3_p\left(\frac{L}{2\pi}\right)^3 \int \frac{\hbar\omega(\mathbf{k})}{e^{\beta\hbar\omega(\mathbf{k})}-1} dk_x dk_y dk_z + E_Z \\
 =&   3_p\left(\frac{L}{2\pi}\right)^3\frac{1}{v_x v_y v_z} \int \frac{\hbar\kappa}{e^{\beta\hbar\kappa} - 1}d\kappa_x d\kappa_y d\kappa_z + E_Z,
@@ -164,8 +164,36 @@ E =& 3_p\left(\frac{L}{2\pi}\right)^3 \int \frac{\hbar\omega(\mathbf{k})}{e^{\be
 where we made the substitutions $\kappa_x = k_x v_x,\kappa_y = k_y v_y, \kappa_z = k_z v_z$ so that $\omega = \kappa$. Going to spherical coordinates:
 
 \begin{align*}
-E &= 3_p \frac{3_p L^3}{2\pi^2}\frac{1}{v_x v_y v_z}\int_{0}^{\kappa_D} \frac{\hbar\kappa^3}{e^{\beta\hbar\kappa} - 1} d\kappa + E_Z  \\
-& = \frac{3_p L^3}{2\pi^2\hbar^3\beta^4}\frac{1}{v_x v_y v_z}\int_0^{\beta\hbar\kappa_D} \frac{x^3}{e^x - 1}dx + E_Z,
+E &= 3_p \frac{3_p L^3}{2\pi^2}\frac{1}{v_x v_y v_z}\int_{0}^{\kappa_D} \frac{\hbar\kappa^3}{e^{\beta\hbar\kappa} - 1} d\kappa + E_Z
+\end{align*}
+
+To calculate the specific heat $C = \frac{dE}{dT}$, let's differentiate this expression directly:
+
+\begin{align*}
+C &= \frac{dE}{dT} = 3_p \frac{3_p L^3}{2\pi^2}\frac{1}{v_x v_y v_z}\int_{0}^{\kappa_D} \frac{d}{dT}\left(\frac{\hbar\kappa^3}{e^{\beta\hbar\kappa} - 1}\right) d\kappa \\
+&= 3_p \frac{3_p L^3}{2\pi^2}\frac{1}{v_x v_y v_z}\int_{0}^{\kappa_D} \frac{\hbar\kappa^3 \cdot \hbar\kappa \cdot e^{\beta\hbar\kappa}}{(e^{\beta\hbar\kappa} - 1)^2} \cdot \frac{d\beta}{dT} d\kappa
+\end{align*}
+
+Since $\beta = \frac{1}{k_B T}$, we have $\frac{d\beta}{dT} = -\frac{1}{k_B T^2} = -\frac{\beta}{T}$. Substituting this:
+
+\begin{align*}
+C &= -3_p \frac{3_p L^3}{2\pi^2}\frac{1}{v_x v_y v_z}\int_{0}^{\kappa_D} \frac{\hbar^2\kappa^4 e^{\beta\hbar\kappa}}{(e^{\beta\hbar\kappa} - 1)^2} \cdot \frac{\beta}{T} d\kappa \\
+&= -3_p \frac{3_p L^3}{2\pi^2}\frac{1}{v_x v_y v_z} \cdot \frac{\hbar^2\beta}{T}\int_{0}^{\kappa_D} \frac{\kappa^4 e^{\beta\hbar\kappa}}{(e^{\beta\hbar\kappa} - 1)^2} d\kappa
+\end{align*}
+
+Now, we can make the substitution $x = \beta\hbar\kappa$, which gives $d\kappa = \frac{dx}{\beta\hbar}$ and changes our limits to $\int_{0}^{\beta\hbar\kappa_D}$:
+
+\begin{align*}
+C &= -3_p \frac{3_p L^3}{2\pi^2}\frac{1}{v_x v_y v_z} \cdot \frac{\hbar^2\beta}{T} \cdot \int_{0}^{\beta\hbar\kappa_D} \left(\frac{x}{\beta\hbar}\right)^4 \frac{e^x}{(e^x - 1)^2} \cdot \frac{dx}{\beta\hbar} \\
+&= -3_p \frac{3_p L^3}{2\pi^2}\frac{1}{v_x v_y v_z} \cdot \frac{\hbar^2\beta}{T} \cdot \frac{1}{(\beta\hbar)^5} \int_{0}^{\beta\hbar\kappa_D} \frac{x^4 e^x}{(e^x - 1)^2} dx \\
+&= -3_p \frac{3_p L^3}{2\pi^2}\frac{1}{v_x v_y v_z} \cdot \frac{1}{T\beta^4\hbar^3} \int_{0}^{\beta\hbar\kappa_D} \frac{x^4 e^x}{(e^x - 1)^2} dx
+\end{align*}
+
+Since $\frac{1}{T\beta^4} = \frac{k_B^4 T^3}{1} = k_B^4 T^3$, we get:
+
+\begin{align*}
+C &= -3_p \frac{3_p L^3}{2\pi^2}\frac{1}{v_x v_y v_z} \cdot \frac{k_B^4 T^3}{\hbar^3} \int_{0}^{\beta\hbar\kappa_D} \frac{x^4 e^x}{(e^x - 1)^2} dx \\
+&= 3_p \frac{3_p L^3 k_B^4 T^3}{2\pi^2\hbar^3}\frac{1}{v_x v_y v_z} \int_{0}^{\beta\hbar\kappa_D} \frac{x^4 e^x}{(e^x - 1)^2} dx
 \end{align*}
 
-Therefore, $C = \frac{dE}{dT} = \frac{6k_B^4 L^3 T^3}{\pi^2\hbar^3}\frac{1}{v_x v_y v_z}\int_0^{\beta\hbar\kappa_D} \frac{x^3}{e^x - 1}dx$. We see that the result is similar to the one with the linear dispersion, the only difference is the factor $1/(v_x v_y v_z)$ instead of $1/v^3$.
+This is similar to the result with isotropic linear dispersion, with the difference being the factor $1/(v_x v_y v_z)$ instead of $1/v^3$.
diff --git a/pixi.lock b/pixi.lock
index c5de99d..8de7cf2 100644
--- a/pixi.lock
+++ b/pixi.lock
@@ -1,4 +1,4 @@
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-  version: '0.1'
-  build: conda_forge
-  subdir: linux-64
-  url: https://conda.anaconda.org/conda-forge/linux-64/_libgcc_mutex-0.1-conda_forge.tar.bz2
+- conda: https://conda.anaconda.org/conda-forge/linux-64/_libgcc_mutex-0.1-conda_forge.tar.bz2
   sha256: fe51de6107f9edc7aa4f786a70f4a883943bc9d39b3bb7307c04c41410990726
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   license: None
   purls: []
   size: 2562
   timestamp: 1578324546067
-- kind: conda
-  name: _openmp_mutex
-  version: '4.5'
-  build: 2_gnu
+- conda: https://conda.anaconda.org/conda-forge/linux-64/_openmp_mutex-4.5-2_gnu.tar.bz2
   build_number: 16
-  subdir: linux-64
-  url: https://conda.anaconda.org/conda-forge/linux-64/_openmp_mutex-4.5-2_gnu.tar.bz2
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@@ -740,12 +730,7 @@ packages:
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   size: 23621
   timestamp: 1650670423406
-- kind: conda
-  name: alsa-lib
-  version: 1.2.12
-  build: h4ab18f5_0
-  subdir: linux-64
-  url: https://conda.anaconda.org/conda-forge/linux-64/alsa-lib-1.2.12-h4ab18f5_0.conda
+- conda: https://conda.anaconda.org/conda-forge/linux-64/alsa-lib-1.2.12-h4ab18f5_0.conda
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@@ -755,12 +740,7 @@ packages:
   purls: []
   size: 555868
   timestamp: 1718118368236
-- kind: conda
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-  version: 3.9.1
-  build: hac33072_0
-  subdir: linux-64
-  url: https://conda.anaconda.org/conda-forge/linux-64/aom-3.9.1-hac33072_0.conda
+- conda: https://conda.anaconda.org/conda-forge/linux-64/aom-3.9.1-hac33072_0.conda
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@@ -771,12 +751,18 @@ packages:
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   size: 2706396
   timestamp: 1718551242397
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-  name: aom
-  version: 3.9.1
-  build: he0c23c2_0
-  subdir: win-64
-  url: https://conda.anaconda.org/conda-forge/win-64/aom-3.9.1-he0c23c2_0.conda
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-  url: https://conda.anaconda.org/conda-forge/osx-64/cairo-1.18.0-h37bd5c4_3.conda
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-  md5: fceaedf1cdbcb02df9699a0d9b005292
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   - zlib
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-  url: https://conda.anaconda.org/conda-forge/noarch/cairocffi-1.6.1-pyhd8ed1ab_0.conda
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-  url: https://conda.anaconda.org/conda-forge/noarch/cairosvg-2.7.1-pyhd8ed1ab_0.conda
+- conda: https://conda.anaconda.org/conda-forge/noarch/cairosvg-2.7.1-pyhd8ed1ab_0.conda
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-  url: https://conda.anaconda.org/conda-forge/noarch/certifi-2024.7.4-pyhd8ed1ab_0.conda
+- conda: https://conda.anaconda.org/conda-forge/noarch/certifi-2024.7.4-pyhd8ed1ab_0.conda
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+- conda: https://conda.anaconda.org/conda-forge/linux-64/cffi-1.17.0-py312h1671c18_0.conda
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+  - ucrt >=10.0.20348.0
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+- conda: https://conda.anaconda.org/conda-forge/noarch/click-8.1.7-unix_pyh707e725_0.conda
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+- conda: https://conda.anaconda.org/conda-forge/noarch/click-8.1.7-win_pyh7428d3b_0.conda
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+- conda: https://conda.anaconda.org/conda-forge/noarch/colorama-0.4.6-pyhd8ed1ab_0.tar.bz2
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+- conda: https://conda.anaconda.org/conda-forge/noarch/comm-0.2.2-pyhd8ed1ab_0.conda
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+- conda: https://conda.anaconda.org/conda-forge/linux-64/contourpy-1.2.1-py312h8572e83_0.conda
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+- conda: https://conda.anaconda.org/conda-forge/osx-64/contourpy-1.2.1-py312h9230928_0.conda
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-  url: https://conda.anaconda.org/conda-forge/noarch/cssselect2-0.2.1-pyh9f0ad1d_1.tar.bz2
+- conda: https://conda.anaconda.org/conda-forge/win-64/contourpy-1.2.1-py312h0d7def4_0.conda
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+  - vc >=14.2,<15
+  - vc14_runtime >=14.29.30139
+  license: BSD-3-Clause
+  license_family: BSD
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+- conda: https://conda.anaconda.org/conda-forge/noarch/cssselect2-0.2.1-pyh9f0ad1d_1.tar.bz2
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-  url: https://conda.anaconda.org/conda-forge/noarch/cycler-0.12.1-pyhd8ed1ab_0.conda
+- conda: https://conda.anaconda.org/conda-forge/noarch/cycler-0.12.1-pyhd8ed1ab_0.conda
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+- conda: https://conda.anaconda.org/conda-forge/linux-64/dav1d-1.2.1-hd590300_0.conda
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+- conda: https://conda.anaconda.org/conda-forge/win-64/dav1d-1.2.1-hcfcfb64_0.conda
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+- conda: https://conda.anaconda.org/conda-forge/linux-64/dbus-1.13.6-h5008d03_3.tar.bz2
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-  url: https://conda.anaconda.org/conda-forge/win-64/zstd-1.5.6-h0ea2cb4_0.conda
-  sha256: 768e30dc513568491818fb068ee867c57c514b553915536da09e5d10b4ebf3c3
-  md5: 9a17230f95733c04dc40a2b1e5491d74
+- conda: https://conda.anaconda.org/conda-forge/linux-64/zstd-1.5.6-ha6fb4c9_0.conda
+  sha256: c558b9cc01d9c1444031bd1ce4b9cff86f9085765f17627a6cd85fc623c8a02b
+  md5: 4d056880988120e29d75bfff282e0f45
   depends:
+  - libgcc-ng >=12
+  - libstdcxx-ng >=12
   - libzlib >=1.2.13,<2.0.0a0
-  - ucrt >=10.0.20348.0
-  - vc >=14.2,<15
-  - vc14_runtime >=14.29.30139
   license: BSD-3-Clause
   license_family: BSD
   purls: []
-  size: 349143
-  timestamp: 1714723445995
-- kind: conda
-  name: zstd
-  version: 1.5.6
-  build: h915ae27_0
-  subdir: osx-64
-  url: https://conda.anaconda.org/conda-forge/osx-64/zstd-1.5.6-h915ae27_0.conda
+  size: 554846
+  timestamp: 1714722996770
+- conda: https://conda.anaconda.org/conda-forge/osx-64/zstd-1.5.6-h915ae27_0.conda
   sha256: efa04a98cb149643fa54c4dad5a0179e36a5fbc88427ea0eec88ceed87fd0f96
   md5: 4cb2cd56f039b129bb0e491c1164167e
   depends:
@@ -10116,20 +7393,16 @@ packages:
   purls: []
   size: 498900
   timestamp: 1714723303098
-- kind: conda
-  name: zstd
-  version: 1.5.6
-  build: ha6fb4c9_0
-  subdir: linux-64
-  url: https://conda.anaconda.org/conda-forge/linux-64/zstd-1.5.6-ha6fb4c9_0.conda
-  sha256: c558b9cc01d9c1444031bd1ce4b9cff86f9085765f17627a6cd85fc623c8a02b
-  md5: 4d056880988120e29d75bfff282e0f45
+- conda: https://conda.anaconda.org/conda-forge/win-64/zstd-1.5.6-h0ea2cb4_0.conda
+  sha256: 768e30dc513568491818fb068ee867c57c514b553915536da09e5d10b4ebf3c3
+  md5: 9a17230f95733c04dc40a2b1e5491d74
   depends:
-  - libgcc-ng >=12
-  - libstdcxx-ng >=12
   - libzlib >=1.2.13,<2.0.0a0
+  - ucrt >=10.0.20348.0
+  - vc >=14.2,<15
+  - vc14_runtime >=14.29.30139
   license: BSD-3-Clause
   license_family: BSD
   purls: []
-  size: 554846
-  timestamp: 1714722996770
+  size: 349143
+  timestamp: 1714723445995
-- 
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