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CHM 421/621 Statistical Mechanics Lecture 4

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Page 1: CHM 421/621 Statistical Mechanicshelios.iiserb.ac.in/~vardha/Courses/CHM421/Lectures/Lec4.pdf · A familiar example Classical microstates - set of instantaneous coordinates and velocities

CHM 421/621 Statistical Mechanics

Lecture 4

Page 2: CHM 421/621 Statistical Mechanicshelios.iiserb.ac.in/~vardha/Courses/CHM421/Lectures/Lec4.pdf · A familiar example Classical microstates - set of instantaneous coordinates and velocities

Introduction and ReviewLecture Plan

An example application of Statistical Mechanics (ctd.): Calculation of average energy

Page 3: CHM 421/621 Statistical Mechanicshelios.iiserb.ac.in/~vardha/Courses/CHM421/Lectures/Lec4.pdf · A familiar example Classical microstates - set of instantaneous coordinates and velocities

A familiar exampleClassical microstates - set of instantaneous coordinates and velocities.

System: Consider a system of N particles moving independently of each other and inside a closed container of volume V. The container (and hence the particles) is assumed to be in thermal contact with a reservoir at temperature T.

Model:

Probability distribution:

Momentum of ith particlePosition of ith particle

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ri<latexit sha1_base64="ZXHsh1bQtmzbwJNJ37CXkDRh7KI=">AAAB8HicbVDLSgNBEOyNrxhfUY9eBoPgKeyKoMegF48RzEOSEGYns8mQmdllplcIS77CiwdFvPo53vwbJ8keNLGgoajqprsrTKSw6PvfXmFtfWNzq7hd2tnd2z8oHx41bZwaxhsslrFph9RyKTRvoEDJ24nhVIWSt8Lx7cxvPXFjRawfcJLwnqJDLSLBKDrpMeuGETHTvuiXK37Vn4OskiAnFchR75e/uoOYpYprZJJa2wn8BHsZNSiY5NNSN7U8oWxMh7zjqKaK2142P3hKzpwyIFFsXGkkc/X3REaVtRMVuk5FcWSXvZn4n9dJMbruZUInKXLNFouiVBKMyex7MhCGM5QTRygzwt1K2IgaytBlVHIhBMsvr5LmRTXwq8H9ZaV2k8dRhBM4hXMI4ApqcAd1aAADBc/wCm+e8V68d+9j0Vrw8plj+APv8wekuJBK</latexit><latexit sha1_base64="ZXHsh1bQtmzbwJNJ37CXkDRh7KI=">AAAB8HicbVDLSgNBEOyNrxhfUY9eBoPgKeyKoMegF48RzEOSEGYns8mQmdllplcIS77CiwdFvPo53vwbJ8keNLGgoajqprsrTKSw6PvfXmFtfWNzq7hd2tnd2z8oHx41bZwaxhsslrFph9RyKTRvoEDJ24nhVIWSt8Lx7cxvPXFjRawfcJLwnqJDLSLBKDrpMeuGETHTvuiXK37Vn4OskiAnFchR75e/uoOYpYprZJJa2wn8BHsZNSiY5NNSN7U8oWxMh7zjqKaK2142P3hKzpwyIFFsXGkkc/X3REaVtRMVuk5FcWSXvZn4n9dJMbruZUInKXLNFouiVBKMyex7MhCGM5QTRygzwt1K2IgaytBlVHIhBMsvr5LmRTXwq8H9ZaV2k8dRhBM4hXMI4ApqcAd1aAADBc/wCm+e8V68d+9j0Vrw8plj+APv8wekuJBK</latexit><latexit sha1_base64="ZXHsh1bQtmzbwJNJ37CXkDRh7KI=">AAAB8HicbVDLSgNBEOyNrxhfUY9eBoPgKeyKoMegF48RzEOSEGYns8mQmdllplcIS77CiwdFvPo53vwbJ8keNLGgoajqprsrTKSw6PvfXmFtfWNzq7hd2tnd2z8oHx41bZwaxhsslrFph9RyKTRvoEDJ24nhVIWSt8Lx7cxvPXFjRawfcJLwnqJDLSLBKDrpMeuGETHTvuiXK37Vn4OskiAnFchR75e/uoOYpYprZJJa2wn8BHsZNSiY5NNSN7U8oWxMh7zjqKaK2142P3hKzpwyIFFsXGkkc/X3REaVtRMVuk5FcWSXvZn4n9dJMbruZUInKXLNFouiVBKMyex7MhCGM5QTRygzwt1K2IgaytBlVHIhBMsvr5LmRTXwq8H9ZaV2k8dRhBM4hXMI4ApqcAd1aAADBc/wCm+e8V68d+9j0Vrw8plj+APv8wekuJBK</latexit><latexit sha1_base64="ZXHsh1bQtmzbwJNJ37CXkDRh7KI=">AAAB8HicbVDLSgNBEOyNrxhfUY9eBoPgKeyKoMegF48RzEOSEGYns8mQmdllplcIS77CiwdFvPo53vwbJ8keNLGgoajqprsrTKSw6PvfXmFtfWNzq7hd2tnd2z8oHx41bZwaxhsslrFph9RyKTRvoEDJ24nhVIWSt8Lx7cxvPXFjRawfcJLwnqJDLSLBKDrpMeuGETHTvuiXK37Vn4OskiAnFchR75e/uoOYpYprZJJa2wn8BHsZNSiY5NNSN7U8oWxMh7zjqKaK2142P3hKzpwyIFFsXGkkc/X3REaVtRMVuk5FcWSXvZn4n9dJMbruZUInKXLNFouiVBKMyex7MhCGM5QTRygzwt1K2IgaytBlVHIhBMsvr5LmRTXwq8H9ZaV2k8dRhBM4hXMI4ApqcAd1aAADBc/wCm+e8V68d+9j0Vrw8plj+APv8wekuJBK</latexit>

� =1

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H({ri(t),pi(t)}) =NX

i=1

1

2mp2i

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⇢({ri,pi}) =1

Q(N,V, T )exp (��H({ri,pi}))

<latexit sha1_base64="TMTWjhkuQZ5ZKkvyPVF46k+fusM=">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</latexit><latexit sha1_base64="TMTWjhkuQZ5ZKkvyPVF46k+fusM=">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</latexit><latexit sha1_base64="TMTWjhkuQZ5ZKkvyPVF46k+fusM=">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</latexit><latexit sha1_base64="TMTWjhkuQZ5ZKkvyPVF46k+fusM=">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</latexit>

Page 4: CHM 421/621 Statistical Mechanicshelios.iiserb.ac.in/~vardha/Courses/CHM421/Lectures/Lec4.pdf · A familiar example Classical microstates - set of instantaneous coordinates and velocities

A familiar exampleProbability distribution:

Meaning: The probability of finding the system in a microstate with the momenta of the particles between pi and pi + dpi and their positions between ri and ri + dri is given by

=>

So what is Q(N,V,T)?

⇢({ri,pi}) =1

Q(N,V, T )exp (��H({ri,pi}))

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⇢ ({ri,pi}) d3Nrid3Npi

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3Npi = 1<latexit sha1_base64="v1/5oNEvw6EHMFyzCdaOcql8Vdc=">AAACRnicbZBNaxRBEIZrNlHj+pGNHr00WYQVZJnRgF4CQS+eJIKbLGyPQ09PzU6Tng+6a4Rl2F/nxXNu/oRcckgQr/bOTiBfBd08/VYV1fXGlVaWfP+P19vYfPDw0dbj/pOnz55vD3ZeHNmyNhInstSlmcbColYFTkiRxmllUOSxxuP45PMqf/wTjVVl8Z0WFYa5mBcqVVKQk6JByFVBjJus5BpTGrU317EREhsep8wsI/WWtVg55EbNM+KmLVg/3rDkR/P+65KZSF1h5XCfBdFg6I/9NthdCDoYQheH0eCUJ6WscyxIamHtLPArChthSEmNyz6vLVZCnog5zhwWIkcbNq0NS/baKQlLS+OOW6pVr3c0Ird2kceuMheU2du5lXhfblZT+jFsVFHVhIVcD0przahkK09ZogxK0gsHQhrl/spkJpxD5JzvOxOC2yvfhaN348AfB9/2hgefOju24BXswggC+AAH8AUOYQISfsEZXMCl99s79/56/9alPa/reQk3ogf/AU9ksOk=</latexit><latexit sha1_base64="v1/5oNEvw6EHMFyzCdaOcql8Vdc=">AAACRnicbZBNaxRBEIZrNlHj+pGNHr00WYQVZJnRgF4CQS+eJIKbLGyPQ09PzU6Tng+6a4Rl2F/nxXNu/oRcckgQr/bOTiBfBd08/VYV1fXGlVaWfP+P19vYfPDw0dbj/pOnz55vD3ZeHNmyNhInstSlmcbColYFTkiRxmllUOSxxuP45PMqf/wTjVVl8Z0WFYa5mBcqVVKQk6JByFVBjJus5BpTGrU317EREhsep8wsI/WWtVg55EbNM+KmLVg/3rDkR/P+65KZSF1h5XCfBdFg6I/9NthdCDoYQheH0eCUJ6WscyxIamHtLPArChthSEmNyz6vLVZCnog5zhwWIkcbNq0NS/baKQlLS+OOW6pVr3c0Ird2kceuMheU2du5lXhfblZT+jFsVFHVhIVcD0przahkK09ZogxK0gsHQhrl/spkJpxD5JzvOxOC2yvfhaN348AfB9/2hgefOju24BXswggC+AAH8AUOYQISfsEZXMCl99s79/56/9alPa/reQk3ogf/AU9ksOk=</latexit><latexit sha1_base64="v1/5oNEvw6EHMFyzCdaOcql8Vdc=">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</latexit><latexit sha1_base64="v1/5oNEvw6EHMFyzCdaOcql8Vdc=">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</latexit>

Page 5: CHM 421/621 Statistical Mechanicshelios.iiserb.ac.in/~vardha/Courses/CHM421/Lectures/Lec4.pdf · A familiar example Classical microstates - set of instantaneous coordinates and velocities

A familiar exampleQ(N,V, T ) = AV N

✓2⇡m

◆ 3N2

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Here A is a constant to ensure the correct dimensions. We will show later that

In this case, the normalisation equation becomes

Q(N,V, T ) =

✓V

⇤3

◆N

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With

A =1

h3N<latexit sha1_base64="8PXXfFi46nz9vdCR3ow7deh7HhU=">AAAB/XicbVDLSsNAFL3xWesrPnZuBovgqiQq6EaounElFewD2lgm00k7dDIJMxOhhuCvuHGhiFv/w51/47TNQlsPXDiccy/33uPHnCntON/W3PzC4tJyYaW4ura+sWlvbddVlEhCayTikWz6WFHOBK1ppjltxpLi0Oe04Q+uRn7jgUrFInGnhzH1QtwTLGAEayN17N0LdI7agcQkdbO0f58e32RZxy45ZWcMNEvcnJQgR7Vjf7W7EUlCKjThWKmW68TaS7HUjHCaFduJojEmA9yjLUMFDqny0vH1GTowShcFkTQlNBqrvydSHCo1DH3TGWLdV9PeSPzPayU6OPNSJuJEU0Emi4KEIx2hURSoyyQlmg8NwUQycysifWyi0CawognBnX55ltSPyq5Tdm9PSpXLPI4C7ME+HIILp1CBa6hCDQg8wjO8wpv1ZL1Y79bHpHXOymd24A+szx9hdJSJ</latexit><latexit sha1_base64="8PXXfFi46nz9vdCR3ow7deh7HhU=">AAAB/XicbVDLSsNAFL3xWesrPnZuBovgqiQq6EaounElFewD2lgm00k7dDIJMxOhhuCvuHGhiFv/w51/47TNQlsPXDiccy/33uPHnCntON/W3PzC4tJyYaW4ura+sWlvbddVlEhCayTikWz6WFHOBK1ppjltxpLi0Oe04Q+uRn7jgUrFInGnhzH1QtwTLGAEayN17N0LdI7agcQkdbO0f58e32RZxy45ZWcMNEvcnJQgR7Vjf7W7EUlCKjThWKmW68TaS7HUjHCaFduJojEmA9yjLUMFDqny0vH1GTowShcFkTQlNBqrvydSHCo1DH3TGWLdV9PeSPzPayU6OPNSJuJEU0Emi4KEIx2hURSoyyQlmg8NwUQycysifWyi0CawognBnX55ltSPyq5Tdm9PSpXLPI4C7ME+HIILp1CBa6hCDQg8wjO8wpv1ZL1Y79bHpHXOymd24A+szx9hdJSJ</latexit><latexit sha1_base64="8PXXfFi46nz9vdCR3ow7deh7HhU=">AAAB/XicbVDLSsNAFL3xWesrPnZuBovgqiQq6EaounElFewD2lgm00k7dDIJMxOhhuCvuHGhiFv/w51/47TNQlsPXDiccy/33uPHnCntON/W3PzC4tJyYaW4ura+sWlvbddVlEhCayTikWz6WFHOBK1ppjltxpLi0Oe04Q+uRn7jgUrFInGnhzH1QtwTLGAEayN17N0LdI7agcQkdbO0f58e32RZxy45ZWcMNEvcnJQgR7Vjf7W7EUlCKjThWKmW68TaS7HUjHCaFduJojEmA9yjLUMFDqny0vH1GTowShcFkTQlNBqrvydSHCo1DH3TGWLdV9PeSPzPayU6OPNSJuJEU0Emi4KEIx2hURSoyyQlmg8NwUQycysifWyi0CawognBnX55ltSPyq5Tdm9PSpXLPI4C7ME+HIILp1CBa6hCDQg8wjO8wpv1ZL1Y79bHpHXOymd24A+szx9hdJSJ</latexit><latexit sha1_base64="8PXXfFi46nz9vdCR3ow7deh7HhU=">AAAB/XicbVDLSsNAFL3xWesrPnZuBovgqiQq6EaounElFewD2lgm00k7dDIJMxOhhuCvuHGhiFv/w51/47TNQlsPXDiccy/33uPHnCntON/W3PzC4tJyYaW4ura+sWlvbddVlEhCayTikWz6WFHOBK1ppjltxpLi0Oe04Q+uRn7jgUrFInGnhzH1QtwTLGAEayN17N0LdI7agcQkdbO0f58e32RZxy45ZWcMNEvcnJQgR7Vjf7W7EUlCKjThWKmW68TaS7HUjHCaFduJojEmA9yjLUMFDqny0vH1GTowShcFkTQlNBqrvydSHCo1DH3TGWLdV9PeSPzPayU6OPNSJuJEU0Emi4KEIx2hURSoyyQlmg8NwUQycysifWyi0CawognBnX55ltSPyq5Tdm9PSpXLPI4C7ME+HIILp1CBa6hCDQg8wjO8wpv1ZL1Y79bHpHXOymd24A+szx9hdJSJ</latexit>

⇤ =hp

2⇡mkBT<latexit sha1_base64="JYzwLOGi1Vy5CQYoCLyOQktiCwE=">AAACEHicbVDLSsNAFJ3UV62vqEs3g0V0VZIi6EYodePCRYW+oAlhMpm0Q2eSODMRSsgnuPFX3LhQxK1Ld/6N0zYLbT0wcDjnXO7c4yeMSmVZ30ZpZXVtfaO8Wdna3tndM/cPujJOBSYdHLNY9H0kCaMR6SiqGOkngiDuM9Lzx9dTv/dAhKRx1FaThLgcDSMaUoyUljzz1LnV4QDBK+iEAuFslGeOvBcqqzsJhRyOvSZs57lnVq2aNQNcJnZBqqBAyzO/nCDGKSeRwgxJObCtRLkZEopiRvKKk0qSIDxGQzLQNEKcSDebHZTDE60EMIyFfpGCM/X3RIa4lBPu6yRHaiQXvan4nzdIVXjpZjRKUkUiPF8UpgyqGE7bgQEVBCs20QRhQfVfIR4h3YvSHVZ0CfbiycukW6/ZVs2+O682mkUdZXAEjsEZsMEFaIAb0AIdgMEjeAav4M14Ml6Md+NjHi0Zxcwh+APj8weaZZxN</latexit><latexit sha1_base64="JYzwLOGi1Vy5CQYoCLyOQktiCwE=">AAACEHicbVDLSsNAFJ3UV62vqEs3g0V0VZIi6EYodePCRYW+oAlhMpm0Q2eSODMRSsgnuPFX3LhQxK1Ld/6N0zYLbT0wcDjnXO7c4yeMSmVZ30ZpZXVtfaO8Wdna3tndM/cPujJOBSYdHLNY9H0kCaMR6SiqGOkngiDuM9Lzx9dTv/dAhKRx1FaThLgcDSMaUoyUljzz1LnV4QDBK+iEAuFslGeOvBcqqzsJhRyOvSZs57lnVq2aNQNcJnZBqqBAyzO/nCDGKSeRwgxJObCtRLkZEopiRvKKk0qSIDxGQzLQNEKcSDebHZTDE60EMIyFfpGCM/X3RIa4lBPu6yRHaiQXvan4nzdIVXjpZjRKUkUiPF8UpgyqGE7bgQEVBCs20QRhQfVfIR4h3YvSHVZ0CfbiycukW6/ZVs2+O682mkUdZXAEjsEZsMEFaIAb0AIdgMEjeAav4M14Ml6Md+NjHi0Zxcwh+APj8weaZZxN</latexit><latexit sha1_base64="JYzwLOGi1Vy5CQYoCLyOQktiCwE=">AAACEHicbVDLSsNAFJ3UV62vqEs3g0V0VZIi6EYodePCRYW+oAlhMpm0Q2eSODMRSsgnuPFX3LhQxK1Ld/6N0zYLbT0wcDjnXO7c4yeMSmVZ30ZpZXVtfaO8Wdna3tndM/cPujJOBSYdHLNY9H0kCaMR6SiqGOkngiDuM9Lzx9dTv/dAhKRx1FaThLgcDSMaUoyUljzz1LnV4QDBK+iEAuFslGeOvBcqqzsJhRyOvSZs57lnVq2aNQNcJnZBqqBAyzO/nCDGKSeRwgxJObCtRLkZEopiRvKKk0qSIDxGQzLQNEKcSDebHZTDE60EMIyFfpGCM/X3RIa4lBPu6yRHaiQXvan4nzdIVXjpZjRKUkUiPF8UpgyqGE7bgQEVBCs20QRhQfVfIR4h3YvSHVZ0CfbiycukW6/ZVs2+O682mkUdZXAEjsEZsMEFaIAb0AIdgMEjeAav4M14Ml6Md+NjHi0Zxcwh+APj8weaZZxN</latexit><latexit sha1_base64="JYzwLOGi1Vy5CQYoCLyOQktiCwE=">AAACEHicbVDLSsNAFJ3UV62vqEs3g0V0VZIi6EYodePCRYW+oAlhMpm0Q2eSODMRSsgnuPFX3LhQxK1Ld/6N0zYLbT0wcDjnXO7c4yeMSmVZ30ZpZXVtfaO8Wdna3tndM/cPujJOBSYdHLNY9H0kCaMR6SiqGOkngiDuM9Lzx9dTv/dAhKRx1FaThLgcDSMaUoyUljzz1LnV4QDBK+iEAuFslGeOvBcqqzsJhRyOvSZs57lnVq2aNQNcJnZBqqBAyzO/nCDGKSeRwgxJObCtRLkZEopiRvKKk0qSIDxGQzLQNEKcSDebHZTDE60EMIyFfpGCM/X3RIa4lBPu6yRHaiQXvan4nzdIVXjpZjRKUkUiPF8UpgyqGE7bgQEVBCs20QRhQfVfIR4h3YvSHVZ0CfbiycukW6/ZVs2+O682mkUdZXAEjsEZsMEFaIAb0AIdgMEjeAav4M14Ml6Md+NjHi0Zxcwh+APj8weaZZxN</latexit>

1

h3N

Z⇢ ({ri,pi}) d3Nrid

3Npi = 1<latexit sha1_base64="Nv6asxcmd7vZVKasPF6uZLXmHew=">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</latexit><latexit sha1_base64="Nv6asxcmd7vZVKasPF6uZLXmHew=">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</latexit><latexit sha1_base64="Nv6asxcmd7vZVKasPF6uZLXmHew=">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</latexit><latexit sha1_base64="Nv6asxcmd7vZVKasPF6uZLXmHew=">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</latexit>

Page 6: CHM 421/621 Statistical Mechanicshelios.iiserb.ac.in/~vardha/Courses/CHM421/Lectures/Lec4.pdf · A familiar example Classical microstates - set of instantaneous coordinates and velocities

Average Energy

U =3

2NkBT

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Calculate

Answer

U = hHi = 1

h3N

Zdr

Zdp H({ri,pi}) ⇢({ri,pi})

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Page 7: CHM 421/621 Statistical Mechanicshelios.iiserb.ac.in/~vardha/Courses/CHM421/Lectures/Lec4.pdf · A familiar example Classical microstates - set of instantaneous coordinates and velocities

Average EnergyU = hHi = 1

h3N

Zdr

Zdp H({ri,pi}) ⇢({ri,pi})

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=1

Qh3N

Zdr

Zdp

NX

i=1

p2i2m

!exp

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NX

i=1

p2i2m

!

<latexit sha1_base64="k4IQXaw+8L0FZmq8GL0uu/gMEj8=">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</latexit><latexit sha1_base64="k4IQXaw+8L0FZmq8GL0uu/gMEj8=">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</latexit><latexit sha1_base64="k4IQXaw+8L0FZmq8GL0uu/gMEj8=">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</latexit><latexit sha1_base64="k4IQXaw+8L0FZmq8GL0uu/gMEj8=">AAACjHiclVFNbxMxEPVu+WgDtKEcuVhESOVA2A2VKEKRqiIhTlUrkbZSnKy8zmxi1d5d2bOIyDJ/hn/Erf8GJ9kDbbkwkqXnN/PGnjd5raTFJLmJ4q0HDx893t7pPHn6bHev+3z/wlaNETASlarMVc4tKFnCCCUquKoNcJ0ruMyvP6/yl9/BWFmV33BZw0TzeSkLKTgGKuv+GlJWGC5c6t05XUzd+1PvKZMl0pljeUHNrVvtfzIFBR4w22impJZoMyeHqZ+eto3qTE4H3g20Z0bOF/iGOmY0hR+130jfshyQ/0eDrNtL+sk66H2QtqBH2jjLur/ZrBKNhhKF4taO06TGieMGpVDgO6yxUHNxzecwDrDkGuzErc309HVgZrSoTDhh7jX7t8Jxbe1S56FSc1zYu7kV+a/cuMHiaOJkWTcIpdg8VDSKYkVXm6EzaUCgWgbAhZHhr1QseHAEw/46wYT07sj3wcWgnyb99Pywd3zS2rFNXpJX5ICk5AM5Jl/JGRkREe1E76Kj6GO8Gx/Gn+LhpjSOWs0LciviL38AnbLG/w==</latexit>

=1

Qh3N

0

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i=1

Z

V

dri

1

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Page 8: CHM 421/621 Statistical Mechanicshelios.iiserb.ac.in/~vardha/Courses/CHM421/Lectures/Lec4.pdf · A familiar example Classical microstates - set of instantaneous coordinates and velocities

Average Energy

Since all pj are being integrated over

each integral in the sum is identical

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Page 9: CHM 421/621 Statistical Mechanicshelios.iiserb.ac.in/~vardha/Courses/CHM421/Lectures/Lec4.pdf · A familiar example Classical microstates - set of instantaneous coordinates and velocities

Average Energy

where we have substituted y =pp2m

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Page 10: CHM 421/621 Statistical Mechanicshelios.iiserb.ac.in/~vardha/Courses/CHM421/Lectures/Lec4.pdf · A familiar example Classical microstates - set of instantaneous coordinates and velocities

Average EnergyNow

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<latexit sha1_base64="f5UkBvQsMuycA7BelCfFyWhcDmk=">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</latexit><latexit sha1_base64="f5UkBvQsMuycA7BelCfFyWhcDmk=">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</latexit><latexit sha1_base64="f5UkBvQsMuycA7BelCfFyWhcDmk=">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</latexit><latexit sha1_base64="f5UkBvQsMuycA7BelCfFyWhcDmk=">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</latexit>

=3⇡

Zdy1 y21exp

���y21

�<latexit sha1_base64="zKvWD7P1qsmC08FluEYtKiuF1Aw=">AAACM3icbVDJSgNBEO1xjXGLevTSGAQ9GGZU0IsgehFPEYwK6Rh6OjVJk56F7hoxDOM3efFHPAjiQRGv/oOd5eD2oODxXhVV9fxESYOu++yMjU9MTk0XZoqzc/MLi6Wl5QsTp1pATcQq1lc+N6BkBDWUqOAq0cBDX8Gl3z3u+5c3oI2Mo3PsJdAIeTuSgRQcrdQsnR5QFmgush2WyDxjPiDPKZMR0hbtNb07W9fbNGM6pHCb5ExBgBtbgz469JiW7Q5u0map7FbcAehf4o1ImYxQbZYeWSsWaQgRCsWNqXtugo2Ma5RCQV5kqYGEiy5vQ93SiIdgGtng55yuW6VFg1jbsscO1O8TGQ+N6YW+7Qw5dsxvry/+59VTDPYbmYySFCESw0VBqijGtB8gbUkNAlXPEi60tLdS0eE2QrQxF20I3u+X/5KL7YrnVryz3fLh0SiOAlkla2SDeGSPHJITUiU1Isg9eSKv5M15cF6cd+dj2DrmjGZWyA84n19KAqlv</latexit><latexit sha1_base64="zKvWD7P1qsmC08FluEYtKiuF1Aw=">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</latexit><latexit sha1_base64="zKvWD7P1qsmC08FluEYtKiuF1Aw=">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</latexit><latexit sha1_base64="zKvWD7P1qsmC08FluEYtKiuF1Aw=">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</latexit>

=3

2�

✓⇡

◆ 32

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Using this formula in the previous equations and setting we get …Q =

✓V

⇤3

◆N

<latexit sha1_base64="lvg2yadY2yki0F7yyTbWTVHOoCU=">AAACD3icbVC7SgNBFJ31GeNr1dJmMCixCbsqaCMEbSxEEjAPyG7C7GQ2GTL7YOauEJb9Axt/xcZCEVtbO//GyaPQxAMDh3Pu4c49Xiy4Asv6NhYWl5ZXVnNr+fWNza1tc2e3rqJEUlajkYhk0yOKCR6yGnAQrBlLRgJPsIY3uB75jQcmFY/CexjGzA1IL+Q+pwS01DGPqvgSO4L5UHR8SWhaz1LnVue7pH2aOZL3+nDcvuuYBatkjYHniT0lBTRFpWN+Od2IJgELgQqiVMu2YnBTIoFTwbK8kygWEzogPdbSNCQBU246vifDh1rpYj+S+oWAx+rvREoCpYaBpycDAn01643E/7xWAv6Fm/IwToCFdLLITwSGCI/KwV0uGQUx1IRQyfVfMe0TXQvoCvO6BHv25HlSPynZVsmunhXKV9M6cmgfHaAistE5KqMbVEE1RNEjekav6M14Ml6Md+NjMrpgTDN76A+Mzx/Ghpvf</latexit><latexit sha1_base64="lvg2yadY2yki0F7yyTbWTVHOoCU=">AAACD3icbVC7SgNBFJ31GeNr1dJmMCixCbsqaCMEbSxEEjAPyG7C7GQ2GTL7YOauEJb9Axt/xcZCEVtbO//GyaPQxAMDh3Pu4c49Xiy4Asv6NhYWl5ZXVnNr+fWNza1tc2e3rqJEUlajkYhk0yOKCR6yGnAQrBlLRgJPsIY3uB75jQcmFY/CexjGzA1IL+Q+pwS01DGPqvgSO4L5UHR8SWhaz1LnVue7pH2aOZL3+nDcvuuYBatkjYHniT0lBTRFpWN+Od2IJgELgQqiVMu2YnBTIoFTwbK8kygWEzogPdbSNCQBU246vifDh1rpYj+S+oWAx+rvREoCpYaBpycDAn01643E/7xWAv6Fm/IwToCFdLLITwSGCI/KwV0uGQUx1IRQyfVfMe0TXQvoCvO6BHv25HlSPynZVsmunhXKV9M6cmgfHaAistE5KqMbVEE1RNEjekav6M14Ml6Md+NjMrpgTDN76A+Mzx/Ghpvf</latexit><latexit sha1_base64="lvg2yadY2yki0F7yyTbWTVHOoCU=">AAACD3icbVC7SgNBFJ31GeNr1dJmMCixCbsqaCMEbSxEEjAPyG7C7GQ2GTL7YOauEJb9Axt/xcZCEVtbO//GyaPQxAMDh3Pu4c49Xiy4Asv6NhYWl5ZXVnNr+fWNza1tc2e3rqJEUlajkYhk0yOKCR6yGnAQrBlLRgJPsIY3uB75jQcmFY/CexjGzA1IL+Q+pwS01DGPqvgSO4L5UHR8SWhaz1LnVue7pH2aOZL3+nDcvuuYBatkjYHniT0lBTRFpWN+Od2IJgELgQqiVMu2YnBTIoFTwbK8kygWEzogPdbSNCQBU246vifDh1rpYj+S+oWAx+rvREoCpYaBpycDAn01643E/7xWAv6Fm/IwToCFdLLITwSGCI/KwV0uGQUx1IRQyfVfMe0TXQvoCvO6BHv25HlSPynZVsmunhXKV9M6cmgfHaAistE5KqMbVEE1RNEjekav6M14Ml6Md+NjMrpgTDN76A+Mzx/Ghpvf</latexit><latexit sha1_base64="lvg2yadY2yki0F7yyTbWTVHOoCU=">AAACD3icbVC7SgNBFJ31GeNr1dJmMCixCbsqaCMEbSxEEjAPyG7C7GQ2GTL7YOauEJb9Axt/xcZCEVtbO//GyaPQxAMDh3Pu4c49Xiy4Asv6NhYWl5ZXVnNr+fWNza1tc2e3rqJEUlajkYhk0yOKCR6yGnAQrBlLRgJPsIY3uB75jQcmFY/CexjGzA1IL+Q+pwS01DGPqvgSO4L5UHR8SWhaz1LnVue7pH2aOZL3+nDcvuuYBatkjYHniT0lBTRFpWN+Od2IJgELgQqiVMu2YnBTIoFTwbK8kygWEzogPdbSNCQBU246vifDh1rpYj+S+oWAx+rvREoCpYaBpycDAn01643E/7xWAv6Fm/IwToCFdLLITwSGCI/KwV0uGQUx1IRQyfVfMe0TXQvoCvO6BHv25HlSPynZVsmunhXKV9M6cmgfHaAistE5KqMbVEE1RNEjekav6M14Ml6Md+NjMrpgTDN76A+Mzx/Ghpvf</latexit>

Page 11: CHM 421/621 Statistical Mechanicshelios.iiserb.ac.in/~vardha/Courses/CHM421/Lectures/Lec4.pdf · A familiar example Classical microstates - set of instantaneous coordinates and velocities

Average Energy

Using

U = N ⇤3N

✓2⇡m

�~2

◆ 3N2 3

2�<latexit sha1_base64="1/qwviFkqgmdx5psPrWUa6ld5dI=">AAACQnicbVDLSiNBFK1WZ0bjqHFcuikMA7oJ3VFwNgPibGYhomDUIZWE25XbSWH1g6rbA6Fpf83NfIE7P8CNC0XcurDyWPiYAwWnzj2HW3XCTCtLvn/jzczOffr8ZX6hsvh1aXmluvrt1Ka5kdiUqU7NeQgWtUqwSYo0nmcGIQ41noUXv0bzs79orEqTExpm2I6hn6hISSAndat/mvwnP7wUBy7Sg06xfVheCo0RbYrIgCwaIlM8LgsRIoEYhGA6jVIY1R/QVmdicZGi4VKTi+Njb9mt1vy6Pwb/SIIpqbEpjrrVa9FLZR5jQlKDta3Az6hdgCElNZYVkVvMQF5AH1uOJhCjbRfjCkr+3Sk9HqXGnYT4WH2dKCC2dhiHzhkDDez72Uj836yVU/SjXagkywkTOVkU5ZpTykd98p4yKEkPHQFplHsrlwNwVZBrveJKCN5/+SM5bdQDvx4c79T29qd1zLN1tsE2WcB22R77zY5Yk0l2xW7ZPXvw/nl33qP3NLHOeNPMGnsD7/kF2DexoA==</latexit><latexit sha1_base64="1/qwviFkqgmdx5psPrWUa6ld5dI=">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</latexit><latexit sha1_base64="1/qwviFkqgmdx5psPrWUa6ld5dI=">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</latexit><latexit sha1_base64="1/qwviFkqgmdx5psPrWUa6ld5dI=">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</latexit>

=3

2NkBT

<latexit sha1_base64="9FcPB1rJa0LBlQpNQ6fGkpkFUEY=">AAAB/XicbVDLSsNAFL2pr1pf8bFzM1gEVyWpgm6EUjeupEJf0IYwmU7aoZMHMxOhhuCvuHGhiFv/w51/47TNQlsPXDiccy/33uPFnEllWd9GYWV1bX2juFna2t7Z3TP3D9oySgShLRLxSHQ9LClnIW0ppjjtxoLiwOO0441vpn7ngQrJorCpJjF1AjwMmc8IVlpyzaPrvi8wSc+ztJqhOzR266jpmmWrYs2AlomdkzLkaLjmV38QkSSgoSIcS9mzrVg5KRaKEU6zUj+RNMZkjIe0p2mIAyqddHZ9hk61MkB+JHSFCs3U3xMpDqScBJ7uDLAayUVvKv7n9RLlXzkpC+NE0ZDMF/kJRypC0yjQgAlKFJ9ogolg+lZERlinoXRgJR2CvfjyMmlXK7ZVse8vyrV6HkcRjuEEzsCGS6jBLTSgBQQe4Rle4c14Ml6Md+Nj3low8plD+APj8wepR5QL</latexit><latexit sha1_base64="9FcPB1rJa0LBlQpNQ6fGkpkFUEY=">AAAB/XicbVDLSsNAFL2pr1pf8bFzM1gEVyWpgm6EUjeupEJf0IYwmU7aoZMHMxOhhuCvuHGhiFv/w51/47TNQlsPXDiccy/33uPFnEllWd9GYWV1bX2juFna2t7Z3TP3D9oySgShLRLxSHQ9LClnIW0ppjjtxoLiwOO0441vpn7ngQrJorCpJjF1AjwMmc8IVlpyzaPrvi8wSc+ztJqhOzR266jpmmWrYs2AlomdkzLkaLjmV38QkSSgoSIcS9mzrVg5KRaKEU6zUj+RNMZkjIe0p2mIAyqddHZ9hk61MkB+JHSFCs3U3xMpDqScBJ7uDLAayUVvKv7n9RLlXzkpC+NE0ZDMF/kJRypC0yjQgAlKFJ9ogolg+lZERlinoXRgJR2CvfjyMmlXK7ZVse8vyrV6HkcRjuEEzsCGS6jBLTSgBQQe4Rle4c14Ml6Md+Nj3low8plD+APj8wepR5QL</latexit><latexit sha1_base64="9FcPB1rJa0LBlQpNQ6fGkpkFUEY=">AAAB/XicbVDLSsNAFL2pr1pf8bFzM1gEVyWpgm6EUjeupEJf0IYwmU7aoZMHMxOhhuCvuHGhiFv/w51/47TNQlsPXDiccy/33uPFnEllWd9GYWV1bX2juFna2t7Z3TP3D9oySgShLRLxSHQ9LClnIW0ppjjtxoLiwOO0441vpn7ngQrJorCpJjF1AjwMmc8IVlpyzaPrvi8wSc+ztJqhOzR266jpmmWrYs2AlomdkzLkaLjmV38QkSSgoSIcS9mzrVg5KRaKEU6zUj+RNMZkjIe0p2mIAyqddHZ9hk61MkB+JHSFCs3U3xMpDqScBJ7uDLAayUVvKv7n9RLlXzkpC+NE0ZDMF/kJRypC0yjQgAlKFJ9ogolg+lZERlinoXRgJR2CvfjyMmlXK7ZVse8vyrV6HkcRjuEEzsCGS6jBLTSgBQQe4Rle4c14Ml6Md+Nj3low8plD+APj8wepR5QL</latexit><latexit sha1_base64="9FcPB1rJa0LBlQpNQ6fGkpkFUEY=">AAAB/XicbVDLSsNAFL2pr1pf8bFzM1gEVyWpgm6EUjeupEJf0IYwmU7aoZMHMxOhhuCvuHGhiFv/w51/47TNQlsPXDiccy/33uPFnEllWd9GYWV1bX2juFna2t7Z3TP3D9oySgShLRLxSHQ9LClnIW0ppjjtxoLiwOO0441vpn7ngQrJorCpJjF1AjwMmc8IVlpyzaPrvi8wSc+ztJqhOzR266jpmmWrYs2AlomdkzLkaLjmV38QkSSgoSIcS9mzrVg5KRaKEU6zUj+RNMZkjIe0p2mIAyqddHZ9hk61MkB+JHSFCs3U3xMpDqScBJ7uDLAayUVvKv7n9RLlXzkpC+NE0ZDMF/kJRypC0yjQgAlKFJ9ogolg+lZERlinoXRgJR2CvfjyMmlXK7ZVse8vyrV6HkcRjuEEzsCGS6jBLTSgBQQe4Rle4c14Ml6Md+Nj3low8plD+APj8wepR5QL</latexit>

� =1

kBT<latexit sha1_base64="DQLS8NEfX2v74sAYQxWlHjx4z0M=">AAACAHicbVBNS8NAEN3Ur1q/oh48eFksgqeSiKAXodSLxwr9giaUzXbSLt1swu5GKCEX/4oXD4p49Wd489+4bXPQ1gcDj/dmmJkXJJwp7TjfVmltfWNzq7xd2dnd2z+wD486Kk4lhTaNeSx7AVHAmYC2ZppDL5FAooBDN5jczfzuI0jFYtHS0wT8iIwECxkl2kgD+8QLQBN8i71QEpq5eTYZNHArH9hVp+bMgVeJW5AqKtAc2F/eMKZpBEJTTpTqu06i/YxIzSiHvOKlChJCJ2QEfUMFiUD52fyBHJ8bZYjDWJoSGs/V3xMZiZSaRoHpjIgeq2VvJv7n9VMd3vgZE0mqQdDFojDlWMd4lgYeMglU86khhEpmbsV0TEwS2mRWMSG4yy+vks5lzXVq7sNVtd4o4iijU3SGLpCLrlEd3aMmaiOKcvSMXtGb9WS9WO/Wx6K1ZBUzx+gPrM8fZFSVnw==</latexit><latexit sha1_base64="DQLS8NEfX2v74sAYQxWlHjx4z0M=">AAACAHicbVBNS8NAEN3Ur1q/oh48eFksgqeSiKAXodSLxwr9giaUzXbSLt1swu5GKCEX/4oXD4p49Wd489+4bXPQ1gcDj/dmmJkXJJwp7TjfVmltfWNzq7xd2dnd2z+wD486Kk4lhTaNeSx7AVHAmYC2ZppDL5FAooBDN5jczfzuI0jFYtHS0wT8iIwECxkl2kgD+8QLQBN8i71QEpq5eTYZNHArH9hVp+bMgVeJW5AqKtAc2F/eMKZpBEJTTpTqu06i/YxIzSiHvOKlChJCJ2QEfUMFiUD52fyBHJ8bZYjDWJoSGs/V3xMZiZSaRoHpjIgeq2VvJv7n9VMd3vgZE0mqQdDFojDlWMd4lgYeMglU86khhEpmbsV0TEwS2mRWMSG4yy+vks5lzXVq7sNVtd4o4iijU3SGLpCLrlEd3aMmaiOKcvSMXtGb9WS9WO/Wx6K1ZBUzx+gPrM8fZFSVnw==</latexit><latexit sha1_base64="DQLS8NEfX2v74sAYQxWlHjx4z0M=">AAACAHicbVBNS8NAEN3Ur1q/oh48eFksgqeSiKAXodSLxwr9giaUzXbSLt1swu5GKCEX/4oXD4p49Wd489+4bXPQ1gcDj/dmmJkXJJwp7TjfVmltfWNzq7xd2dnd2z+wD486Kk4lhTaNeSx7AVHAmYC2ZppDL5FAooBDN5jczfzuI0jFYtHS0wT8iIwECxkl2kgD+8QLQBN8i71QEpq5eTYZNHArH9hVp+bMgVeJW5AqKtAc2F/eMKZpBEJTTpTqu06i/YxIzSiHvOKlChJCJ2QEfUMFiUD52fyBHJ8bZYjDWJoSGs/V3xMZiZSaRoHpjIgeq2VvJv7n9VMd3vgZE0mqQdDFojDlWMd4lgYeMglU86khhEpmbsV0TEwS2mRWMSG4yy+vks5lzXVq7sNVtd4o4iijU3SGLpCLrlEd3aMmaiOKcvSMXtGb9WS9WO/Wx6K1ZBUzx+gPrM8fZFSVnw==</latexit><latexit sha1_base64="DQLS8NEfX2v74sAYQxWlHjx4z0M=">AAACAHicbVBNS8NAEN3Ur1q/oh48eFksgqeSiKAXodSLxwr9giaUzXbSLt1swu5GKCEX/4oXD4p49Wd489+4bXPQ1gcDj/dmmJkXJJwp7TjfVmltfWNzq7xd2dnd2z+wD486Kk4lhTaNeSx7AVHAmYC2ZppDL5FAooBDN5jczfzuI0jFYtHS0wT8iIwECxkl2kgD+8QLQBN8i71QEpq5eTYZNHArH9hVp+bMgVeJW5AqKtAc2F/eMKZpBEJTTpTqu06i/YxIzSiHvOKlChJCJ2QEfUMFiUD52fyBHJ8bZYjDWJoSGs/V3xMZiZSaRoHpjIgeq2VvJv7n9VMd3vgZE0mqQdDFojDlWMd4lgYeMglU86khhEpmbsV0TEwS2mRWMSG4yy+vks5lzXVq7sNVtd4o4iijU3SGLpCLrlEd3aMmaiOKcvSMXtGb9WS9WO/Wx6K1ZBUzx+gPrM8fZFSVnw==</latexit>

Which is the desired result!

⇤ =hp

2⇡mkBT<latexit sha1_base64="JYzwLOGi1Vy5CQYoCLyOQktiCwE=">AAACEHicbVDLSsNAFJ3UV62vqEs3g0V0VZIi6EYodePCRYW+oAlhMpm0Q2eSODMRSsgnuPFX3LhQxK1Ld/6N0zYLbT0wcDjnXO7c4yeMSmVZ30ZpZXVtfaO8Wdna3tndM/cPujJOBSYdHLNY9H0kCaMR6SiqGOkngiDuM9Lzx9dTv/dAhKRx1FaThLgcDSMaUoyUljzz1LnV4QDBK+iEAuFslGeOvBcqqzsJhRyOvSZs57lnVq2aNQNcJnZBqqBAyzO/nCDGKSeRwgxJObCtRLkZEopiRvKKk0qSIDxGQzLQNEKcSDebHZTDE60EMIyFfpGCM/X3RIa4lBPu6yRHaiQXvan4nzdIVXjpZjRKUkUiPF8UpgyqGE7bgQEVBCs20QRhQfVfIR4h3YvSHVZ0CfbiycukW6/ZVs2+O682mkUdZXAEjsEZsMEFaIAb0AIdgMEjeAav4M14Ml6Md+NjHi0Zxcwh+APj8weaZZxN</latexit><latexit sha1_base64="JYzwLOGi1Vy5CQYoCLyOQktiCwE=">AAACEHicbVDLSsNAFJ3UV62vqEs3g0V0VZIi6EYodePCRYW+oAlhMpm0Q2eSODMRSsgnuPFX3LhQxK1Ld/6N0zYLbT0wcDjnXO7c4yeMSmVZ30ZpZXVtfaO8Wdna3tndM/cPujJOBSYdHLNY9H0kCaMR6SiqGOkngiDuM9Lzx9dTv/dAhKRx1FaThLgcDSMaUoyUljzz1LnV4QDBK+iEAuFslGeOvBcqqzsJhRyOvSZs57lnVq2aNQNcJnZBqqBAyzO/nCDGKSeRwgxJObCtRLkZEopiRvKKk0qSIDxGQzLQNEKcSDebHZTDE60EMIyFfpGCM/X3RIa4lBPu6yRHaiQXvan4nzdIVXjpZjRKUkUiPF8UpgyqGE7bgQEVBCs20QRhQfVfIR4h3YvSHVZ0CfbiycukW6/ZVs2+O682mkUdZXAEjsEZsMEFaIAb0AIdgMEjeAav4M14Ml6Md+NjHi0Zxcwh+APj8weaZZxN</latexit><latexit sha1_base64="JYzwLOGi1Vy5CQYoCLyOQktiCwE=">AAACEHicbVDLSsNAFJ3UV62vqEs3g0V0VZIi6EYodePCRYW+oAlhMpm0Q2eSODMRSsgnuPFX3LhQxK1Ld/6N0zYLbT0wcDjnXO7c4yeMSmVZ30ZpZXVtfaO8Wdna3tndM/cPujJOBSYdHLNY9H0kCaMR6SiqGOkngiDuM9Lzx9dTv/dAhKRx1FaThLgcDSMaUoyUljzz1LnV4QDBK+iEAuFslGeOvBcqqzsJhRyOvSZs57lnVq2aNQNcJnZBqqBAyzO/nCDGKSeRwgxJObCtRLkZEopiRvKKk0qSIDxGQzLQNEKcSDebHZTDE60EMIyFfpGCM/X3RIa4lBPu6yRHaiQXvan4nzdIVXjpZjRKUkUiPF8UpgyqGE7bgQEVBCs20QRhQfVfIR4h3YvSHVZ0CfbiycukW6/ZVs2+O682mkUdZXAEjsEZsMEFaIAb0AIdgMEjeAav4M14Ml6Md+NjHi0Zxcwh+APj8weaZZxN</latexit><latexit sha1_base64="JYzwLOGi1Vy5CQYoCLyOQktiCwE=">AAACEHicbVDLSsNAFJ3UV62vqEs3g0V0VZIi6EYodePCRYW+oAlhMpm0Q2eSODMRSsgnuPFX3LhQxK1Ld/6N0zYLbT0wcDjnXO7c4yeMSmVZ30ZpZXVtfaO8Wdna3tndM/cPujJOBSYdHLNY9H0kCaMR6SiqGOkngiDuM9Lzx9dTv/dAhKRx1FaThLgcDSMaUoyUljzz1LnV4QDBK+iEAuFslGeOvBcqqzsJhRyOvSZs57lnVq2aNQNcJnZBqqBAyzO/nCDGKSeRwgxJObCtRLkZEopiRvKKk0qSIDxGQzLQNEKcSDebHZTDE60EMIyFfpGCM/X3RIa4lBPu6yRHaiQXvan4nzdIVXjpZjRKUkUiPF8UpgyqGE7bgQEVBCs20QRhQfVfIR4h3YvSHVZ0CfbiycukW6/ZVs2+O682mkUdZXAEjsEZsMEFaIAb0AIdgMEjeAav4M14Ml6Md+NjHi0Zxcwh+APj8weaZZxN</latexit>