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SEO Keyword summary for en.wikipedia.org/wiki/equipartition_theorem
Keywords are extracted from the main content of your website and are the primary indicator of the words this page could rank for. By frequenty count we expect your focus keyword to be energy
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Short Tail Keywords energy frac equipartition |
long Tail Keywords (2 words) equipartition theorem kinetic energy heat capacity ideal gas partial hpartial |
long Tail Keywords (3 words) degree of freedom ideal gas law hmathrm kin partial partial hmathrm kin translational kinetic energy average kinetic energy hmathrm pot rangle |
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equipartition theorem wikipedia
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theorem found in path !
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https://wikimedia.org/api/rest_v1/media/math/render/svg/23a7c0289eb6c3229b42fcc618f0452ea1bc9bf3 height: height attribute not set width: width attribute not set description: {\displaystyle p\oint _{\mathrm {surface} }\mathbf {q} \cdot \mathbf {ds} =p\int _{\mathrm {volume} }\left({\boldsymbol {\nabla }}\cdot \mathbf {q} \right)\,dv=3pv,} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/d06d5bd7a0cc196b5b77c156df7be4479dac4701 height: height attribute not set width: width attribute not set description: {\displaystyle 3nk_{\text{b}}t=-\left\langle \sum _{k=1}^{n}\mathbf {q} _{k}\cdot \mathbf {f} _{k}\right\rangle =3pv,} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/cb4459a59c1ccd72f82be16b49e4782aec6b0c81 height: height attribute not set width: width attribute not set description: {\displaystyle pv=nk_{\text{b}}t=nrt,} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/a11e702ce3b4d7792549399685a8da6f644f5a96 height: height attribute not set width: width attribute not set description: {\displaystyle h={\frac {\left|\mathbf {p} _{1}\right|^{2}}{2m_{1}}}+{\frac {\left|\mathbf {p} _{2}\right|^{2}}{2m_{2}}}+{\frac {1}{2}}aq^{2},} |
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https://upload.wikimedia.org/wikipedia/commons/thumb/c/c9/chandra-crab.jpg/280px-chandra-crab.jpg height: 280 width: 280 description: no alt description found |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/94bbb9b2737858292c6950fb12d152e2b07b6b84 height: height attribute not set width: width attribute not set description: {\displaystyle h_{\mathrm {kin} }\approx cp=c{\sqrt {p_{x}^{2}+p_{y}^{2}+p_{z}^{2}}}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/9cf14effcaf68249ae6bc59fad0f4144ab1eb682 height: height attribute not set width: width attribute not set description: {\displaystyle p_{x}{\frac {\partial h_{\mathrm {kin} }}{\partial p_{x}}}=c{\frac {p_{x}^{2}}{\sqrt {p_{x}^{2}+p_{y}^{2}+p_{z}^{2}}}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/31f01af86080a2f3dbd21b0818ed33bbb2dc2b77 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}\langle h_{\mathrm {kin} }\rangle &=\left\langle c{\frac {p_{x}^{2}+p_{y}^{2}+p_{z}^{2}}{\sqrt {p_{x}^{2}+p_{y}^{2}+p_{z}^{2}}}}\right\rangle \\&=\left\langle p_{x}{\frac {\partial h^{\mathrm {kin} }}{\partial p_{x}}}\right\rangle +\left\langle p_{y}{\frac {\partial h^{\mathrm {kin} }}{\partial p_{y}}}\right\rangle +\left\langle p_{z}{\frac {\partial h^{\mathrm {kin} }}{\partial p_{z}}}\right\rangle \\&=3k_{\text{b}}t\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/bbd8ca8c03292686243d097751c6e02c7bcfa518 height: height attribute not set width: width attribute not set description: {\displaystyle \langle h_{\mathrm {pot} }\rangle =\int _{0}^{\infty }4\pi r^{2}\rho u(r)g(r)\,dr.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/9b18551c4a91f25cd18e80c1bc28702fb6cee19a height: height attribute not set width: width attribute not set description: {\displaystyle \langle h_{\text{pot}}\rangle ={\tfrac {1}{2}}n\langle h_{\mathrm {pot} }\rangle } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/83ee5bedf2c710e0bb24980028c8a4cb31aff27a height: height attribute not set width: width attribute not set description: {\displaystyle h=\langle h_{\mathrm {kin} }\rangle +\langle h_{\mathrm {pot} }\rangle ={\frac {3}{2}}nk_{\text{b}}t+2\pi n\rho \int _{0}^{\infty }r^{2}u(r)g(r)\,dr.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/a577dc31563046c4eb99dee8edfdb8d7e2121159 height: height attribute not set width: width attribute not set description: {\displaystyle 3nk_{\text{b}}t=3pv+2\pi n\rho \int _{0}^{\infty }r^{3}u'(r)g(r)\,dr.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/c9e1d9280a68aef4b3266c17fcdbb7d7dd5da69c height: height attribute not set width: width attribute not set description: {\displaystyle h_{\mathrm {pot} }=cq^{s},\,} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/6a8f0e23dceb56a80e111539c1d9f3bec446ca2a height: height attribute not set width: width attribute not set description: {\displaystyle k_{\text{b}}t=\left\langle q{\frac {\partial h_{\mathrm {pot} }}{\partial q}}\right\rangle =\langle q\cdot scq^{s-1}\rangle =\langle scq^{s}\rangle =s\langle h_{\mathrm {pot} }\rangle .} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/6bee61138b8574b1fcfa6cbb5024cbfc7e0f6ff5 height: height attribute not set width: width attribute not set description: {\displaystyle h_{\mathrm {pot} }=\sum _{n=2}^{\infty }c_{n}q^{n}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/e3cf005fa2878cafa6211eba8e522fcd448f637c height: height attribute not set width: width attribute not set description: {\displaystyle k_{\text{b}}t=\left\langle q{\frac {\partial h_{\mathrm {pot} }}{\partial q}}\right\rangle =\sum _{n=2}^{\infty }\langle q\cdot nc_{n}q^{n-1}\rangle =\sum _{n=2}^{\infty }nc_{n}\langle q^{n}\rangle .} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/10ebcb29ac6d79f3064e15926b13a289d6ea4e1f height: height attribute not set width: width attribute not set description: {\displaystyle \langle h_{\mathrm {pot} }\rangle ={\frac {1}{2}}k_{\text{b}}t-\sum _{n=3}^{\infty }\left({\frac {n-2}{2}}\right)c_{n}\langle q^{n}\rangle } |
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https://upload.wikimedia.org/wikipedia/commons/thumb/f/f8/wiener_process_3d.png/330px-wiener_process_3d.png height: 322 width: 330 description: no alt description found |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/63b698f245197aa236121978bd6bc02073c6542e height: height attribute not set width: width attribute not set description: {\displaystyle {\frac {d\mathbf {v} }{dt}}={\frac {1}{m}}\mathbf {f} =-{\frac {\mathbf {v} }{\tau }}+{\frac {1}{m}}\mathbf {f} _{\mathrm {rnd} },} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/fb71d6b8d415b20f5901acdc580d3a4d34de4e3e height: height attribute not set width: width attribute not set description: {\displaystyle \left\langle \mathbf {r} \cdot {\frac {d\mathbf {v} }{dt}}\right\rangle +{\frac {1}{\tau }}\langle \mathbf {r} \cdot \mathbf {v} \rangle =0} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/bf8f094b6015c8def8328e3d263dd55797cc5912 height: height attribute not set width: width attribute not set description: {\displaystyle {\frac {d}{dt}}\left(\mathbf {r} \cdot \mathbf {r} \right)={\frac {d}{dt}}\left(r^{2}\right)=2\left(\mathbf {r} \cdot \mathbf {v} \right)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/0240aa15099e4ccf0f2562c2b9136739e29d8815 height: height attribute not set width: width attribute not set description: {\displaystyle {\frac {d}{dt}}\left(\mathbf {r} \cdot \mathbf {v} \right)=v^{2}+\mathbf {r} \cdot {\frac {d\mathbf {v} }{dt}},} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/2bfcbd0ac962026759708c1a0440788fcfc25465 height: height attribute not set width: width attribute not set description: {\displaystyle {\frac {d^{2}}{dt^{2}}}\langle r^{2}\rangle +{\frac {1}{\tau }}{\frac {d}{dt}}\langle r^{2}\rangle =2\langle v^{2}\rangle ={\frac {6}{m}}k_{\text{b}}t,} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/93d213980f00ffb510beefa2b4b581a66072c7d2 height: height attribute not set width: width attribute not set description: {\displaystyle \langle h_{\mathrm {kin} }\rangle =\left\langle {\frac {p^{2}}{2m}}\right\rangle =\langle {\tfrac {1}{2}}mv^{2}\rangle ={\tfrac {3}{2}}k_{\text{b}}t.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/089684b1048796cfa989cc0ac044571b77d3017a height: height attribute not set width: width attribute not set description: {\displaystyle \langle r^{2}\rangle } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/d2ff145e424c0f0101b793a8b88718e28f1d6b3c height: height attribute not set width: width attribute not set description: {\displaystyle \langle r^{2}\rangle ={\frac {6k_{\text{b}}t\tau ^{2}}{m}}\left(e^{-t/\tau }-1+{\frac {t}{\tau }}\right).} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/b49885103fad931f8a0787a1c8fadb8dddd7b5a8 height: height attribute not set width: width attribute not set description: {\displaystyle \langle r^{2}\rangle \approx {\frac {3k_{\text{b}}t}{m}}t^{2}=\langle v^{2}\rangle t^{2}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/56976576ae52063ae9a44db2cb09e35abb0b499b height: height attribute not set width: width attribute not set description: {\displaystyle \langle r^{2}\rangle \approx {\frac {6k_{\text{b}}t\tau }{m}}t={\frac {6k_{\text{b}}tt}{\gamma }}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/09575b27b0bba70237e2c215e39abe9540a6fd29 height: height attribute not set width: width attribute not set description: {\displaystyle h_{\mathrm {grav} }=-\int _{0}^{r}{\frac {4\pi r^{2}g}{r}}m(r)\,\rho (r)\,dr,} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/7d6c8cf1ee80a6d2068b6d1c5c71f4ab5c37e7ab height: height attribute not set width: width attribute not set description: {\displaystyle h_{\mathrm {grav} }=-{\frac {3gm^{2}}{5r}},} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/afa4c9b8d44dba0177017cf393aaf38532c25e9b height: height attribute not set width: width attribute not set description: {\displaystyle \langle h_{\mathrm {grav} }\rangle ={\frac {h_{\mathrm {grav} }}{n}}=-{\frac {3gm^{2}}{5rn}},} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/b201acd00f0a47fe778092cba0e49c1b0a85ffb8 height: height attribute not set width: width attribute not set description: {\displaystyle \left\langle r{\frac {\partial h_{\mathrm {grav} }}{\partial r}}\right\rangle =\langle -h_{\mathrm {grav} }\rangle =k_{\text{b}}t={\frac {3gm^{2}}{5rn}}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/500d1b088a652bfd2627ecd0e25ed654a237d1ec height: height attribute not set width: width attribute not set description: {\displaystyle {\frac {3gm^{2}}{5r}}>3nk_{\text{b}}t.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/4b2b4c920d251b6a77afe5a9ad4c0e15300bc5b1 height: height attribute not set width: width attribute not set description: {\displaystyle m={\frac {4}{3}}\pi r^{3}\rho } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/0386c7268430d968c279b085e055088a34044a47 height: height attribute not set width: width attribute not set description: {\displaystyle m_{\text{j}}^{2}=\left({\frac {5k_{\text{b}}t}{gm_{p}}}\right)^{3}\left({\frac {3}{4\pi \rho }}\right).} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/3c4f6330b6a2370becc1a7dccd53e127878ecef5 height: height attribute not set width: width attribute not set description: {\displaystyle f(v_{z})={\sqrt {\dfrac {m}{2\pi k_{\text{b}}t}}}\;e^{\frac {-m{v_{z}}^{2}}{2k_{\text{b}}t}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/96d6f476fc9d1b4f307642e752d55c58882015e6 height: height attribute not set width: width attribute not set description: {\displaystyle \langle {v_{z}}^{2}\rangle =\int _{-\infty }^{\infty }f(v_{z}){v_{z}}^{2}dv_{z}={\dfrac {k_{\text{b}}t}{m}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/9e8f9947a2217ef8c2aafbad91cbb66365a94069 height: height attribute not set width: width attribute not set description: {\displaystyle \langle e_{k}\rangle ={\dfrac {3}{2}}m\langle {v_{z}}^{2}\rangle ={\dfrac {3}{2}}k_{\text{b}}t} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/ca160a4f51f9a50f8883beda24bb870fbf6203f9 height: height attribute not set width: width attribute not set description: {\displaystyle z_{x}=\int _{-\infty }^{\infty }dx\ e^{-\beta ax^{2}}={\sqrt {\frac {\pi }{\beta a}}},} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/00b52e2d4625bb3bb0ef16d305d6124e3a319143 height: height attribute not set width: width attribute not set description: {\displaystyle \langle h_{x}\rangle =-{\frac {\partial \log z_{x}}{\partial \beta }}={\frac {1}{2\beta }}={\frac {1}{2}}k_{\text{b}}t} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/e8727c2f215e5e314487d84f6c0b47e6b6680970 height: height attribute not set width: width attribute not set description: {\displaystyle d\gamma =\prod _{i}dq_{i}\,dp_{i}\,} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/1d6046bc798a322d5b4f20954cd5aea15a9f77e4 height: height attribute not set width: width attribute not set description: {\displaystyle \sigma (e,\delta e)=\int _{h\in \left[e,e+\delta e\right]}d\gamma .} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/d8b9ea97663d10ff04a1553ed79476e36ab02744 height: height attribute not set width: width attribute not set description: {\displaystyle \omega (e)=\int _{h |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/dacf6f8ccd4ccf04ffa00683a62e9327772e5f0f height: height attribute not set width: width attribute not set description: {\displaystyle \int _{h\in \left[e,e+\delta e\right]}\ldots d\gamma =\delta e{\frac {\partial }{\partial e}}\int _{h |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/953f69473c41ba21fa77f79e5e887edf02435c79 height: height attribute not set width: width attribute not set description: {\displaystyle \sigma (e,\delta e)=\delta e\ {\frac {\partial \omega }{\partial e}}=\delta e\ \rho (e),} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/4f5d899ad89128130e440fd61141e70d56f6b6d3 height: height attribute not set width: width attribute not set description: {\displaystyle {\frac {1}{t}}={\frac {\partial s}{\partial e}}=k_{\text{b}}{\frac {\partial \log \omega }{\partial e}}=k_{\text{b}}{\frac {1}{\omega }}\,{\frac {\partial \omega }{\partial e}}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/b7551c7bed2cd2ee83e10536d157c94a5f8f72fd height: height attribute not set width: width attribute not set description: {\displaystyle {\mathcal {n}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/7f760ff7da808f5e9d44cb2a95d6e3885c5b3e15 height: height attribute not set width: width attribute not set description: {\displaystyle {\mathcal {n}}\int e^{-\beta h(p,q)}d\gamma =1,} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/3580cb598b3bd491034528a2db5456cc0a2f440e height: height attribute not set width: width attribute not set description: {\displaystyle {\mathcal {n}}\int e^{-\beta h(p,q)}d\gamma ={\mathcal {n}}\int d[x_{k}e^{-\beta h(p,q)}]d\gamma _{k}-{\mathcal {n}}\int x_{k}{\frac {\partial e^{-\beta h(p,q)}}{\partial x_{k}}}d\gamma ,} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/4e941a978fe3d56ab99364c732bd077bb21eee1e height: height attribute not set width: width attribute not set description: {\displaystyle {\mathcal {n}}\int \left[e^{-\beta h(p,q)}x_{k}\right]_{x_{k}=a}^{x_{k}=b}d\gamma _{k}+{\mathcal {n}}\int e^{-\beta h(p,q)}x_{k}\beta {\frac {\partial h}{\partial x_{k}}}d\gamma =1,} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/5e00b454d8db32e5e8df1a40a1516896c5c7f332 height: height attribute not set width: width attribute not set description: {\displaystyle {\mathcal {n}}\int e^{-\beta h(p,q)}x_{k}{\frac {\partial h}{\partial x_{k}}}\,d\gamma =\left\langle x_{k}{\frac {\partial h}{\partial x_{k}}}\right\rangle ={\frac {1}{\beta }}=k_{\text{b}}t.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/f8f365afc3303fd926e94a51545a0d72d54a2eb5 height: height attribute not set width: width attribute not set description: {\displaystyle \langle \ldots \rangle } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/bab05fb103d0b069f3eb97aa28ab9885a0578796 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}\left\langle x_{m}{\frac {\partial h}{\partial x_{n}}}\right\rangle &={\frac {1}{\sigma }}\,\int _{h\in \left[e,e+\delta e\right]}x_{m}{\frac {\partial h}{\partial x_{n}}}\,d\gamma \\&={\frac {\delta e}{\sigma }}\,{\frac {\partial }{\partial e}}\int _{h |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/801cfe2696ae376a32f3dd8fe04fcd7bc7741cab height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}\int _{h |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/8e6f657a99b975d5cc98b9d97481f863e839a91d height: height attribute not set width: width attribute not set description: {\displaystyle \left\langle x_{m}{\frac {\partial h}{\partial x_{n}}}\right\rangle =\delta _{mn}{\frac {1}{\rho }}\,{\frac {\partial }{\partial e}}\int _{h |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/2444985f873bd92b54ba7b4ae87aeaf284eec88a height: height attribute not set width: width attribute not set description: {\displaystyle \rho ={\frac {\partial \omega }{\partial e}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/b6d2e6e00fa802a9aad468497ca9d6e55ec5a87e height: height attribute not set width: width attribute not set description: {\displaystyle \left\langle x_{m}{\frac {\partial h}{\partial x_{n}}}\right\rangle =\delta _{mn}\left({\frac {1}{\omega }}{\frac {\partial \omega }{\partial e}}\right)^{-1}=\delta _{mn}\left({\frac {\partial \log \omega }{\partial e}}\right)^{-1}=\delta _{mn}k_{\text{b}}t.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/a3273049809dcc645e0179b576b7df1fce39f44e height: height attribute not set width: width attribute not set description: {\displaystyle \left\langle x_{m}{\frac {\partial h}{\partial x_{n}}}\right\rangle =\delta _{mn}k_{\text{b}}t,} |
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https://upload.wikimedia.org/wikipedia/commons/thumb/5/57/et_fig2.png/320px-et_fig2.png height: 307 width: 320 description: no alt description found |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/d99e63227fd20235e75040135ef627d983650610 height: height attribute not set width: width attribute not set description: {\displaystyle p(e_{n})={\frac {e^{-n\beta h\nu }}{z}},} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/d1733c494cd72b32f736ebbbe2f3795b8c932da6 height: height attribute not set width: width attribute not set description: {\displaystyle z=\sum _{n=0}^{\infty }e^{-n\beta h\nu }={\frac {1}{1-e^{-\beta h\nu }}}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/d2fc69d830484a1930d4dc0eeba446ab6adbc086 height: height attribute not set width: width attribute not set description: {\displaystyle \langle h\rangle =\sum _{n=0}^{\infty }e_{n}p(e_{n})={\frac {1}{z}}\sum _{n=0}^{\infty }nh\nu \ e^{-n\beta h\nu }=-{\frac {1}{z}}{\frac {\partial z}{\partial \beta }}=-{\frac {\partial \log z}{\partial \beta }}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/fa1cd7ecada61be8ef2f89d5c40a4c3ee9f637ee height: height attribute not set width: width attribute not set description: {\displaystyle \langle h\rangle =h\nu {\frac {e^{-\beta h\nu }}{1-e^{-\beta h\nu }}}.} |
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