Cenni di meccanica statistica - 1 - Termodinamica - Corso B

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Stefano Olivares Quantum Technology Lab Dipartimento di Fisica “Aldo Pontremoli” Università degli Studi di Milano, Italy “Aldo Pontremoli” Cenni di meccanica statistica — 1 Termodinamica - Corso B [email protected] http://users.unimi.it/aqm https://solivarest.ariel.ctu.unimi.it Termodinamica — a.a. 2019-2020

Transcript of Cenni di meccanica statistica - 1 - Termodinamica - Corso B

Stefano Olivares Quantum Technology Lab Dipartimento di Fisica “Aldo Pontremoli” Università degli Studi di Milano, Italy

“Aldo Pontremoli”

Cenni di meccanica statistica — 1 Termodinamica - Corso B

[email protected]://users.unimi.it/aqm

https://solivarest.ariel.ctu.unimi.it

Termodinamica — a.a. 2019-2020

Assegnate le condizioni fisiche esiste una partizione più probabile, raggiunta la quale il sistema risulta in equilibrio (può fluttuare attorno a questa senza cambiarla).

Il problema della meccanica statistica è trovare questa partizione.

Qui consideriamo un sistema isolato formato da un gran numero N di particelle, che possono distribuirsi su vari livelli energetici.

Per un sistema isolato l’energia totale U è costante, ma può variare la distribuzione (partizione) tra i vari livelli energetici.

La meccanica statisticaLa meccanica statistica applica i principi della meccanica classica a sistemi di moltissime particelle per ottenere le proprietà macroscopiche del sistema.

La statistica di Maxwell—Boltzmann

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Se abbiamo N particelle identiche e distinguibili:

W =N !

n1! (N � n1)!| {z }W1

(N � n1)!

n2! (N � n1 � n2)!| {z }W2

· · ·

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numero di modi in cui posso scegliere tra N particelle le che occupano n1 E1

Assumiamo, per ora, che i livelli energetici abbiano uguale probabilità di essere occupati (no degenerazione).

La probabilità di una particolare partizione è proporzionale al numero W di modi diversi in cui si possono distribuire le particelle tra

{n1, n2, …}

E1, E2, …

Consideriamo un sistema isolato formato da un gran numero N di particelle.

Assumiamo, per ora, che i livelli energetici abbiano uguale probabilità di essere occupati (no degenerazione).

La probabilità di una particolare partizione è proporzionale al numero W di modi diversi in cui si possono distribuire le particelle tra

{n1, n2, …}

E1, E2, …

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Se abbiamo N particelle identiche e distinguibili:

La statistica di Maxwell—Boltzmann

Consideriamo un sistema isolato formato da un gran numero N di particelle.

W = N !Y

i

1

ni!<latexit sha1_base64="jrfuspiZXyAsnBu4a677UqB3r2s=">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</latexit><latexit sha1_base64="jrfuspiZXyAsnBu4a677UqB3r2s=">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</latexit><latexit sha1_base64="jrfuspiZXyAsnBu4a677UqB3r2s=">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</latexit>

Se il livello ha una degenerazione , le particelle possono occupare uno qualsiasi di questi livelli.

Ei gi ni

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n2<latexit sha1_base64="gXHS5zmVWA/MH4ihOdYHRSLQaaw=">AAAB6XicdVBNS8NAEJ34WetX1aOXxSJ4CklbbXMrePFY0dhCG8pmu22XbjZhdyOU0J/gxYOKV/+RN/+Nm7aCij4YeLw3w8y8MOFMacf5sFZW19Y3Ngtbxe2d3b390sHhnYpTSahPYh7LTogV5UxQXzPNaSeRFEchp+1wcpn77XsqFYvFrZ4mNIjwSLAhI1gb6Ub0K/1S2bFrVa/iXiBDalW3Ws/JueM1HOTazhxlWKLVL733BjFJIyo04VipruskOsiw1IxwOiv2UkUTTCZ4RLuGChxRFWTzU2fo1CgDNIylKaHRXP0+keFIqWkUms4I67H67eXiX1431cNGkDGRpJoKslg0TDnSMcr/RgMmKdF8aggmkplbERljiYk26RRNCF+fov+JX7E927mulZveMo0CHMMJnIELdWjCFbTABwIjeIAneLa49Wi9WK+L1hVrOXMEP2C9fQLSEY2q</latexit><latexit sha1_base64="gXHS5zmVWA/MH4ihOdYHRSLQaaw=">AAAB6XicdVBNS8NAEJ34WetX1aOXxSJ4CklbbXMrePFY0dhCG8pmu22XbjZhdyOU0J/gxYOKV/+RN/+Nm7aCij4YeLw3w8y8MOFMacf5sFZW19Y3Ngtbxe2d3b390sHhnYpTSahPYh7LTogV5UxQXzPNaSeRFEchp+1wcpn77XsqFYvFrZ4mNIjwSLAhI1gb6Ub0K/1S2bFrVa/iXiBDalW3Ws/JueM1HOTazhxlWKLVL733BjFJIyo04VipruskOsiw1IxwOiv2UkUTTCZ4RLuGChxRFWTzU2fo1CgDNIylKaHRXP0+keFIqWkUms4I67H67eXiX1431cNGkDGRpJoKslg0TDnSMcr/RgMmKdF8aggmkplbERljiYk26RRNCF+fov+JX7E927mulZveMo0CHMMJnIELdWjCFbTABwIjeIAneLa49Wi9WK+L1hVrOXMEP2C9fQLSEY2q</latexit><latexit sha1_base64="gXHS5zmVWA/MH4ihOdYHRSLQaaw=">AAAB6XicdVBNS8NAEJ34WetX1aOXxSJ4CklbbXMrePFY0dhCG8pmu22XbjZhdyOU0J/gxYOKV/+RN/+Nm7aCij4YeLw3w8y8MOFMacf5sFZW19Y3Ngtbxe2d3b390sHhnYpTSahPYh7LTogV5UxQXzPNaSeRFEchp+1wcpn77XsqFYvFrZ4mNIjwSLAhI1gb6Ub0K/1S2bFrVa/iXiBDalW3Ws/JueM1HOTazhxlWKLVL733BjFJIyo04VipruskOsiw1IxwOiv2UkUTTCZ4RLuGChxRFWTzU2fo1CgDNIylKaHRXP0+keFIqWkUms4I67H67eXiX1431cNGkDGRpJoKslg0TDnSMcr/RgMmKdF8aggmkplbERljiYk26RRNCF+fov+JX7E927mulZveMo0CHMMJnIELdWjCFbTABwIjeIAneLa49Wi9WK+L1hVrOXMEP2C9fQLSEY2q</latexit>

ni<latexit sha1_base64="2TsETHPB09/fdJFNCwp4Zjlex/s=">AAAB6XicdVBNS8NAEJ34WetX1aOXxSJ4CklbbXMrePFY0dhCG8pmu2mXbjZhdyOU0p/gxYOKV/+RN/+Nm7aCij4YeLw3w8y8MOVMacf5sFZW19Y3Ngtbxe2d3b390sHhnUoySahPEp7ITogV5UxQXzPNaSeVFMchp+1wfJn77XsqFUvErZ6kNIjxULCIEayNdCP6rF8qO3at6lXcC2RIrepW6zk5d7yGg1zbmaMMS7T6pffeICFZTIUmHCvVdZ1UB1MsNSOczoq9TNEUkzEe0q6hAsdUBdP5qTN0apQBihJpSmg0V79PTHGs1CQOTWeM9Uj99nLxL6+b6agRTJlIM00FWSyKMo50gvK/0YBJSjSfGIKJZOZWREZYYqJNOkUTwten6H/iV2zPdq5r5aa3TKMAx3ACZ+BCHZpwBS3wgcAQHuAJni1uPVov1uuidcVazhzBD1hvnyVFjeE=</latexit><latexit sha1_base64="2TsETHPB09/fdJFNCwp4Zjlex/s=">AAAB6XicdVBNS8NAEJ34WetX1aOXxSJ4CklbbXMrePFY0dhCG8pmu2mXbjZhdyOU0p/gxYOKV/+RN/+Nm7aCij4YeLw3w8y8MOVMacf5sFZW19Y3Ngtbxe2d3b390sHhnUoySahPEp7ITogV5UxQXzPNaSeVFMchp+1wfJn77XsqFUvErZ6kNIjxULCIEayNdCP6rF8qO3at6lXcC2RIrepW6zk5d7yGg1zbmaMMS7T6pffeICFZTIUmHCvVdZ1UB1MsNSOczoq9TNEUkzEe0q6hAsdUBdP5qTN0apQBihJpSmg0V79PTHGs1CQOTWeM9Uj99nLxL6+b6agRTJlIM00FWSyKMo50gvK/0YBJSjSfGIKJZOZWREZYYqJNOkUTwten6H/iV2zPdq5r5aa3TKMAx3ACZ+BCHZpwBS3wgcAQHuAJni1uPVov1uuidcVazhzBD1hvnyVFjeE=</latexit><latexit sha1_base64="2TsETHPB09/fdJFNCwp4Zjlex/s=">AAAB6XicdVBNS8NAEJ34WetX1aOXxSJ4CklbbXMrePFY0dhCG8pmu2mXbjZhdyOU0p/gxYOKV/+RN/+Nm7aCij4YeLw3w8y8MOVMacf5sFZW19Y3Ngtbxe2d3b390sHhnUoySahPEp7ITogV5UxQXzPNaSeVFMchp+1wfJn77XsqFUvErZ6kNIjxULCIEayNdCP6rF8qO3at6lXcC2RIrepW6zk5d7yGg1zbmaMMS7T6pffeICFZTIUmHCvVdZ1UB1MsNSOczoq9TNEUkzEe0q6hAsdUBdP5qTN0apQBihJpSmg0V79PTHGs1CQOTWeM9Uj99nLxL6+b6agRTJlIM00FWSyKMo50gvK/0YBJSjSfGIKJZOZWREZYYqJNOkUTwten6H/iV2zPdq5r5aa3TKMAx3ACZ+BCHZpwBS3wgcAQHuAJni1uPVov1uuidcVazhzBD1hvnyVFjeE=</latexit>

La statistica di Maxwell—Boltzmann

g1<latexit sha1_base64="LQ3N1OCSNUNRezAzjJ8x0a6/ltY=">AAAB6XicdVBNS8NAEN34WetX1aOXxSJ4Cpu22uZW8OKxorGFNpTNdpMu3WzC7kYooT/BiwcVr/4jb/4bN20FFX0w8Hhvhpl5QcqZ0gh9WCura+sbm6Wt8vbO7t5+5eDwTiWZJNQjCU9kL8CKciaop5nmtJdKiuOA024wuSz87j2ViiXiVk9T6sc4EixkBGsj3URDZ1ipIrtRd2vOBTSkUXfqzYKcI7eFoGOjOapgic6w8j4YJSSLqdCEY6X6Dkq1n2OpGeF0Vh5kiqaYTHBE+4YKHFPl5/NTZ/DUKCMYJtKU0HCufp/IcazUNA5MZ4z1WP32CvEvr5/psOXnTKSZpoIsFoUZhzqBxd9wxCQlmk8NwUQycyskYywx0Sadsgnh61P4P/Fqtmuj60a17S7TKIFjcALOgAOaoA2uQAd4gIAIPIAn8Gxx69F6sV4XrSvWcuYI/ID19gnF642i</latexit><latexit sha1_base64="LQ3N1OCSNUNRezAzjJ8x0a6/ltY=">AAAB6XicdVBNS8NAEN34WetX1aOXxSJ4Cpu22uZW8OKxorGFNpTNdpMu3WzC7kYooT/BiwcVr/4jb/4bN20FFX0w8Hhvhpl5QcqZ0gh9WCura+sbm6Wt8vbO7t5+5eDwTiWZJNQjCU9kL8CKciaop5nmtJdKiuOA024wuSz87j2ViiXiVk9T6sc4EixkBGsj3URDZ1ipIrtRd2vOBTSkUXfqzYKcI7eFoGOjOapgic6w8j4YJSSLqdCEY6X6Dkq1n2OpGeF0Vh5kiqaYTHBE+4YKHFPl5/NTZ/DUKCMYJtKU0HCufp/IcazUNA5MZ4z1WP32CvEvr5/psOXnTKSZpoIsFoUZhzqBxd9wxCQlmk8NwUQycyskYywx0Sadsgnh61P4P/Fqtmuj60a17S7TKIFjcALOgAOaoA2uQAd4gIAIPIAn8Gxx69F6sV4XrSvWcuYI/ID19gnF642i</latexit><latexit sha1_base64="LQ3N1OCSNUNRezAzjJ8x0a6/ltY=">AAAB6XicdVBNS8NAEN34WetX1aOXxSJ4Cpu22uZW8OKxorGFNpTNdpMu3WzC7kYooT/BiwcVr/4jb/4bN20FFX0w8Hhvhpl5QcqZ0gh9WCura+sbm6Wt8vbO7t5+5eDwTiWZJNQjCU9kL8CKciaop5nmtJdKiuOA024wuSz87j2ViiXiVk9T6sc4EixkBGsj3URDZ1ipIrtRd2vOBTSkUXfqzYKcI7eFoGOjOapgic6w8j4YJSSLqdCEY6X6Dkq1n2OpGeF0Vh5kiqaYTHBE+4YKHFPl5/NTZ/DUKCMYJtKU0HCufp/IcazUNA5MZ4z1WP32CvEvr5/psOXnTKSZpoIsFoUZhzqBxd9wxCQlmk8NwUQycyskYywx0Sadsgnh61P4P/Fqtmuj60a17S7TKIFjcALOgAOaoA2uQAd4gIAIPIAn8Gxx69F6sV4XrSvWcuYI/ID19gnF642i</latexit>

g2<latexit sha1_base64="rZQjqZwAzGpZfhqyj33Vl7EVQg8=">AAAB6XicdVBNS8NAEN34WetX1aOXxSJ4Cpu22uZW8OKxorGFNpTNdpMu3WzC7kYooT/BiwcVr/4jb/4bN20FFX0w8Hhvhpl5QcqZ0gh9WCura+sbm6Wt8vbO7t5+5eDwTiWZJNQjCU9kL8CKciaop5nmtJdKiuOA024wuSz87j2ViiXiVk9T6sc4EixkBGsj3UTD2rBSRXaj7tacC2hIo+7UmwU5R24LQcdGc1TBEp1h5X0wSkgWU6EJx0r1HZRqP8dSM8LprDzIFE0xmeCI9g0VOKbKz+enzuCpUUYwTKQpoeFc/T6R41ipaRyYzhjrsfrtFeJfXj/TYcvPmUgzTQVZLAozDnUCi7/hiElKNJ8agolk5lZIxlhiok06ZRPC16fwf+LVbNdG141q212mUQLH4AScAQc0QRtcgQ7wAAEReABP4Nni1qP1Yr0uWles5cwR+AHr7RPHbo2j</latexit><latexit sha1_base64="rZQjqZwAzGpZfhqyj33Vl7EVQg8=">AAAB6XicdVBNS8NAEN34WetX1aOXxSJ4Cpu22uZW8OKxorGFNpTNdpMu3WzC7kYooT/BiwcVr/4jb/4bN20FFX0w8Hhvhpl5QcqZ0gh9WCura+sbm6Wt8vbO7t5+5eDwTiWZJNQjCU9kL8CKciaop5nmtJdKiuOA024wuSz87j2ViiXiVk9T6sc4EixkBGsj3UTD2rBSRXaj7tacC2hIo+7UmwU5R24LQcdGc1TBEp1h5X0wSkgWU6EJx0r1HZRqP8dSM8LprDzIFE0xmeCI9g0VOKbKz+enzuCpUUYwTKQpoeFc/T6R41ipaRyYzhjrsfrtFeJfXj/TYcvPmUgzTQVZLAozDnUCi7/hiElKNJ8agolk5lZIxlhiok06ZRPC16fwf+LVbNdG141q212mUQLH4AScAQc0QRtcgQ7wAAEReABP4Nni1qP1Yr0uWles5cwR+AHr7RPHbo2j</latexit><latexit sha1_base64="rZQjqZwAzGpZfhqyj33Vl7EVQg8=">AAAB6XicdVBNS8NAEN34WetX1aOXxSJ4Cpu22uZW8OKxorGFNpTNdpMu3WzC7kYooT/BiwcVr/4jb/4bN20FFX0w8Hhvhpl5QcqZ0gh9WCura+sbm6Wt8vbO7t5+5eDwTiWZJNQjCU9kL8CKciaop5nmtJdKiuOA024wuSz87j2ViiXiVk9T6sc4EixkBGsj3UTD2rBSRXaj7tacC2hIo+7UmwU5R24LQcdGc1TBEp1h5X0wSkgWU6EJx0r1HZRqP8dSM8LprDzIFE0xmeCI9g0VOKbKz+enzuCpUUYwTKQpoeFc/T6R41ipaRyYzhjrsfrtFeJfXj/TYcvPmUgzTQVZLAozDnUCi7/hiElKNJ8agolk5lZIxlhiok06ZRPC16fwf+LVbNdG141q212mUQLH4AScAQc0QRtcgQ7wAAEReABP4Nni1qP1Yr0uWles5cwR+AHr7RPHbo2j</latexit>

gi<latexit sha1_base64="NvsM2T/CqZFWR+IOB/1j+VGwRzI=">AAAB6XicdVBNS8NAEN34WetX1aOXxSJ4Cpu22uZW8OKxorGFNpTNdpMu3WzC7kYooT/BiwcVr/4jb/4bN20FFX0w8Hhvhpl5QcqZ0gh9WCura+sbm6Wt8vbO7t5+5eDwTiWZJNQjCU9kL8CKciaop5nmtJdKiuOA024wuSz87j2ViiXiVk9T6sc4EixkBGsj3URDNqxUkd2ouzXnAhrSqDv1ZkHOkdtC0LHRHFWwRGdYeR+MEpLFVGjCsVJ9B6Xaz7HUjHA6Kw8yRVNMJjiifUMFjqny8/mpM3hqlBEME2lKaDhXv0/kOFZqGgemM8Z6rH57hfiX18902PJzJtJMU0EWi8KMQ53A4m84YpISzaeGYCKZuRWSMZaYaJNO2YTw9Sn8n3g127XRdaPadpdplMAxOAFnwAFN0AZXoAM8QEAEHsATeLa49Wi9WK+L1hVrOXMEfsB6+wQaoo3a</latexit><latexit sha1_base64="NvsM2T/CqZFWR+IOB/1j+VGwRzI=">AAAB6XicdVBNS8NAEN34WetX1aOXxSJ4Cpu22uZW8OKxorGFNpTNdpMu3WzC7kYooT/BiwcVr/4jb/4bN20FFX0w8Hhvhpl5QcqZ0gh9WCura+sbm6Wt8vbO7t5+5eDwTiWZJNQjCU9kL8CKciaop5nmtJdKiuOA024wuSz87j2ViiXiVk9T6sc4EixkBGsj3URDNqxUkd2ouzXnAhrSqDv1ZkHOkdtC0LHRHFWwRGdYeR+MEpLFVGjCsVJ9B6Xaz7HUjHA6Kw8yRVNMJjiifUMFjqny8/mpM3hqlBEME2lKaDhXv0/kOFZqGgemM8Z6rH57hfiX18902PJzJtJMU0EWi8KMQ53A4m84YpISzaeGYCKZuRWSMZaYaJNO2YTw9Sn8n3g127XRdaPadpdplMAxOAFnwAFN0AZXoAM8QEAEHsATeLa49Wi9WK+L1hVrOXMEfsB6+wQaoo3a</latexit><latexit sha1_base64="NvsM2T/CqZFWR+IOB/1j+VGwRzI=">AAAB6XicdVBNS8NAEN34WetX1aOXxSJ4Cpu22uZW8OKxorGFNpTNdpMu3WzC7kYooT/BiwcVr/4jb/4bN20FFX0w8Hhvhpl5QcqZ0gh9WCura+sbm6Wt8vbO7t5+5eDwTiWZJNQjCU9kL8CKciaop5nmtJdKiuOA024wuSz87j2ViiXiVk9T6sc4EixkBGsj3URDNqxUkd2ouzXnAhrSqDv1ZkHOkdtC0LHRHFWwRGdYeR+MEpLFVGjCsVJ9B6Xaz7HUjHA6Kw8yRVNMJjiifUMFjqny8/mpM3hqlBEME2lKaDhXv0/kOFZqGgemM8Z6rH57hfiX18902PJzJtJMU0EWi8KMQ53A4m84YpISzaeGYCKZuRWSMZaYaJNO2YTw9Sn8n3g127XRdaPadpdplMAxOAFnwAFN0AZXoAM8QEAEHsATeLa49Wi9WK+L1hVrOXMEfsB6+wQaoo3a</latexit>

La probabilità di una particolare partizione è data ora da:{n1, n2, …}

W = N !Y

i

gnii

ni!<latexit sha1_base64="q6yeXsuDR0vyNCcOluT6zF8yO2E=">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</latexit><latexit sha1_base64="q6yeXsuDR0vyNCcOluT6zF8yO2E=">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</latexit><latexit sha1_base64="q6yeXsuDR0vyNCcOluT6zF8yO2E=">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</latexit>

NOTA — Il fatto che le particelle siano assunte distinguibili porta a risultati non fisici come il cosiddetto paradosso di Gibbs.

Consideriamo un sistema isolato formato da un gran numero N di particelle.

Se il livello ha una degenerazione , le particelle possono occupare uno qualsiasi di questi livelli.

Ei gi ni

La probabilità di una particolare partizione è data ora da:{n1, n2, …}

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E2<latexit sha1_base64="3RQPJ3NAIFIrzzfuwzW+pkmmsrY=">AAAB6nicdVBNSwMxEM36WetX1aOXYBE8Ldm22u6tIILHCq4ttEvJpmkbmmSXJCuUpX/BiwcVr/4ib/4bs20FFX0w8Hhvhpl5UcKZNgh9OCura+sbm4Wt4vbO7t5+6eDwTsepIjQgMY9VJ8KaciZpYJjhtJMoikXEaTuaXOZ++54qzWJ5a6YJDQUeSTZkBJtcuupXiv1SGbm1ql/xLqAltapXrefkHPkNBD0XzVEGS7T6pffeICapoNIQjrXueigxYYaVYYTTWbGXappgMsEj2rVUYkF1mM1vncFTqwzgMFa2pIFz9ftEhoXWUxHZToHNWP/2cvEvr5uaYSPMmExSQyVZLBqmHJoY5o/DAVOUGD61BBPF7K2QjLHCxNh48hC+PoX/k6Di+i66qZWb/jKNAjgGJ+AMeKAOmuAatEAACBiDB/AEnh3hPDovzuuidcVZzhyBH3DePgHIn42V</latexit><latexit sha1_base64="3RQPJ3NAIFIrzzfuwzW+pkmmsrY=">AAAB6nicdVBNSwMxEM36WetX1aOXYBE8Ldm22u6tIILHCq4ttEvJpmkbmmSXJCuUpX/BiwcVr/4ib/4bs20FFX0w8Hhvhpl5UcKZNgh9OCura+sbm4Wt4vbO7t5+6eDwTsepIjQgMY9VJ8KaciZpYJjhtJMoikXEaTuaXOZ++54qzWJ5a6YJDQUeSTZkBJtcuupXiv1SGbm1ql/xLqAltapXrefkHPkNBD0XzVEGS7T6pffeICapoNIQjrXueigxYYaVYYTTWbGXappgMsEj2rVUYkF1mM1vncFTqwzgMFa2pIFz9ftEhoXWUxHZToHNWP/2cvEvr5uaYSPMmExSQyVZLBqmHJoY5o/DAVOUGD61BBPF7K2QjLHCxNh48hC+PoX/k6Di+i66qZWb/jKNAjgGJ+AMeKAOmuAatEAACBiDB/AEnh3hPDovzuuidcVZzhyBH3DePgHIn42V</latexit><latexit sha1_base64="3RQPJ3NAIFIrzzfuwzW+pkmmsrY=">AAAB6nicdVBNSwMxEM36WetX1aOXYBE8Ldm22u6tIILHCq4ttEvJpmkbmmSXJCuUpX/BiwcVr/4ib/4bs20FFX0w8Hhvhpl5UcKZNgh9OCura+sbm4Wt4vbO7t5+6eDwTsepIjQgMY9VJ8KaciZpYJjhtJMoikXEaTuaXOZ++54qzWJ5a6YJDQUeSTZkBJtcuupXiv1SGbm1ql/xLqAltapXrefkHPkNBD0XzVEGS7T6pffeICapoNIQjrXueigxYYaVYYTTWbGXappgMsEj2rVUYkF1mM1vncFTqwzgMFa2pIFz9ftEhoXWUxHZToHNWP/2cvEvr5uaYSPMmExSQyVZLBqmHJoY5o/DAVOUGD61BBPF7K2QjLHCxNh48hC+PoX/k6Di+i66qZWb/jKNAjgGJ+AMeKAOmuAatEAACBiDB/AEnh3hPDovzuuidcVZzhyBH3DePgHIn42V</latexit>

Ei<latexit sha1_base64="7X0umSOWrNb2X3PvRXE11IukntI=">AAAB6nicdVBNSwMxEM36WetX1aOXYBE8Ldm22u6tIILHCq4ttEvJptk2NMkuSVYopX/BiwcVr/4ib/4bs20FFX0w8Hhvhpl5UcqZNgh9OCura+sbm4Wt4vbO7t5+6eDwTieZIjQgCU9UJ8KaciZpYJjhtJMqikXEaTsaX+Z++54qzRJ5ayYpDQUeShYzgk0uXfVZsV8qI7dW9SveBbSkVvWq9ZycI7+BoOeiOcpgiVa/9N4bJCQTVBrCsdZdD6UmnGJlGOF0VuxlmqaYjPGQdi2VWFAdTue3zuCpVQYwTpQtaeBc/T4xxULriYhsp8BmpH97ufiX181M3AinTKaZoZIsFsUZhyaB+eNwwBQlhk8swUQxeyskI6wwMTaePISvT+H/JKi4votuauWmv0yjAI7BCTgDHqiDJrgGLRAAAkbgATyBZ0c4j86L87poXXGWM0fgB5y3TxwKjcw=</latexit><latexit sha1_base64="7X0umSOWrNb2X3PvRXE11IukntI=">AAAB6nicdVBNSwMxEM36WetX1aOXYBE8Ldm22u6tIILHCq4ttEvJptk2NMkuSVYopX/BiwcVr/4ib/4bs20FFX0w8Hhvhpl5UcqZNgh9OCura+sbm4Wt4vbO7t5+6eDwTieZIjQgCU9UJ8KaciZpYJjhtJMqikXEaTsaX+Z++54qzRJ5ayYpDQUeShYzgk0uXfVZsV8qI7dW9SveBbSkVvWq9ZycI7+BoOeiOcpgiVa/9N4bJCQTVBrCsdZdD6UmnGJlGOF0VuxlmqaYjPGQdi2VWFAdTue3zuCpVQYwTpQtaeBc/T4xxULriYhsp8BmpH97ufiX181M3AinTKaZoZIsFsUZhyaB+eNwwBQlhk8swUQxeyskI6wwMTaePISvT+H/JKi4votuauWmv0yjAI7BCTgDHqiDJrgGLRAAAkbgATyBZ0c4j86L87poXXGWM0fgB5y3TxwKjcw=</latexit><latexit sha1_base64="7X0umSOWrNb2X3PvRXE11IukntI=">AAAB6nicdVBNSwMxEM36WetX1aOXYBE8Ldm22u6tIILHCq4ttEvJptk2NMkuSVYopX/BiwcVr/4ib/4bs20FFX0w8Hhvhpl5UcqZNgh9OCura+sbm4Wt4vbO7t5+6eDwTieZIjQgCU9UJ8KaciZpYJjhtJMqikXEaTsaX+Z++54qzRJ5ayYpDQUeShYzgk0uXfVZsV8qI7dW9SveBbSkVvWq9ZycI7+BoOeiOcpgiVa/9N4bJCQTVBrCsdZdD6UmnGJlGOF0VuxlmqaYjPGQdi2VWFAdTue3zuCpVQYwTpQtaeBc/T4xxULriYhsp8BmpH97ufiX181M3AinTKaZoZIsFsUZhyaB+eNwwBQlhk8swUQxeyskI6wwMTaePISvT+H/JKi4votuauWmv0yjAI7BCTgDHqiDJrgGLRAAAkbgATyBZ0c4j86L87poXXGWM0fgB5y3TxwKjcw=</latexit>

...<latexit sha1_base64="ZK6F/zo8fNhL8QtCjZJg7pU98D4=">AAAB7XicdVBNS8NAEN3Ur1q/qh69LBbBU0jaatuLFLx4rGBsoQ1ls9m0Sze7YXdTKKE/wosHFa/+H2/+GzdtBRV9MPB4b4aZeUHCqNKO82EV1tY3NreK26Wd3b39g/Lh0b0SqcTEw4IJ2QuQIoxy4mmqGeklkqA4YKQbTK5zvzslUlHB7/QsIX6MRpxGFCNtpO5gGgqtSsNyxbHrtVbVvYSG1GturZGTC6fVdKBrOwtUwAqdYfl9EAqcxoRrzJBSfddJtJ8hqSlmZF4apIokCE/QiPQN5Sgmys8W587hmVFCGAlpimu4UL9PZChWahYHpjNGeqx+e7n4l9dPddT0M8qTVBOOl4uilEEtYP47DKkkWLOZIQhLam6FeIwkwtoklIfw9Sn8n3hVu2U7t/VK+2qVRhGcgFNwDlzQAG1wAzrAAxhMwAN4As9WYj1aL9brsrVgrWaOwQ9Yb5/WH49u</latexit><latexit sha1_base64="ZK6F/zo8fNhL8QtCjZJg7pU98D4=">AAAB7XicdVBNS8NAEN3Ur1q/qh69LBbBU0jaatuLFLx4rGBsoQ1ls9m0Sze7YXdTKKE/wosHFa/+H2/+GzdtBRV9MPB4b4aZeUHCqNKO82EV1tY3NreK26Wd3b39g/Lh0b0SqcTEw4IJ2QuQIoxy4mmqGeklkqA4YKQbTK5zvzslUlHB7/QsIX6MRpxGFCNtpO5gGgqtSsNyxbHrtVbVvYSG1GturZGTC6fVdKBrOwtUwAqdYfl9EAqcxoRrzJBSfddJtJ8hqSlmZF4apIokCE/QiPQN5Sgmys8W587hmVFCGAlpimu4UL9PZChWahYHpjNGeqx+e7n4l9dPddT0M8qTVBOOl4uilEEtYP47DKkkWLOZIQhLam6FeIwkwtoklIfw9Sn8n3hVu2U7t/VK+2qVRhGcgFNwDlzQAG1wAzrAAxhMwAN4As9WYj1aL9brsrVgrWaOwQ9Yb5/WH49u</latexit><latexit sha1_base64="ZK6F/zo8fNhL8QtCjZJg7pU98D4=">AAAB7XicdVBNS8NAEN3Ur1q/qh69LBbBU0jaatuLFLx4rGBsoQ1ls9m0Sze7YXdTKKE/wosHFa/+H2/+GzdtBRV9MPB4b4aZeUHCqNKO82EV1tY3NreK26Wd3b39g/Lh0b0SqcTEw4IJ2QuQIoxy4mmqGeklkqA4YKQbTK5zvzslUlHB7/QsIX6MRpxGFCNtpO5gGgqtSsNyxbHrtVbVvYSG1GturZGTC6fVdKBrOwtUwAqdYfl9EAqcxoRrzJBSfddJtJ8hqSlmZF4apIokCE/QiPQN5Sgmys8W587hmVFCGAlpimu4UL9PZChWahYHpjNGeqx+e7n4l9dPddT0M8qTVBOOl4uilEEtYP47DKkkWLOZIQhLam6FeIwkwtoklIfw9Sn8n3hVu2U7t/VK+2qVRhGcgFNwDlzQAG1wAzrAAxhMwAN4As9WYj1aL9brsrVgrWaOwQ9Yb5/WH49u</latexit>

n1<latexit sha1_base64="A0GRrpinw0p56xt5t36SJXA6dAI=">AAAB6XicdVBNS8NAEJ34WetX1aOXxSJ4CklbbXMrePFY0dhCG8pmu2mXbjZhdyOU0p/gxYOKV/+RN/+Nm7aCij4YeLw3w8y8MOVMacf5sFZW19Y3Ngtbxe2d3b390sHhnUoySahPEp7ITogV5UxQXzPNaSeVFMchp+1wfJn77XsqFUvErZ6kNIjxULCIEayNdCP6br9Uduxa1au4F8iQWtWt1nNy7ngNB7m2M0cZlmj1S++9QUKymApNOFaq6zqpDqZYakY4nRV7maIpJmM8pF1DBY6pCqbzU2fo1CgDFCXSlNBorn6fmOJYqUkcms4Y65H67eXiX14301EjmDKRZpoKslgUZRzpBOV/owGTlGg+MQQTycytiIywxESbdIomhK9P0f/Er9ie7VzXyk1vmUYBjuEEzsCFOjThClrgA4EhPMATPFvcerRerNdF64q1nDmCH7DePgHQjo2p</latexit><latexit sha1_base64="A0GRrpinw0p56xt5t36SJXA6dAI=">AAAB6XicdVBNS8NAEJ34WetX1aOXxSJ4CklbbXMrePFY0dhCG8pmu2mXbjZhdyOU0p/gxYOKV/+RN/+Nm7aCij4YeLw3w8y8MOVMacf5sFZW19Y3Ngtbxe2d3b390sHhnUoySahPEp7ITogV5UxQXzPNaSeVFMchp+1wfJn77XsqFUvErZ6kNIjxULCIEayNdCP6br9Uduxa1au4F8iQWtWt1nNy7ngNB7m2M0cZlmj1S++9QUKymApNOFaq6zqpDqZYakY4nRV7maIpJmM8pF1DBY6pCqbzU2fo1CgDFCXSlNBorn6fmOJYqUkcms4Y65H67eXiX14301EjmDKRZpoKslgUZRzpBOV/owGTlGg+MQQTycytiIywxESbdIomhK9P0f/Er9ie7VzXyk1vmUYBjuEEzsCFOjThClrgA4EhPMATPFvcerRerNdF64q1nDmCH7DePgHQjo2p</latexit><latexit sha1_base64="A0GRrpinw0p56xt5t36SJXA6dAI=">AAAB6XicdVBNS8NAEJ34WetX1aOXxSJ4CklbbXMrePFY0dhCG8pmu2mXbjZhdyOU0p/gxYOKV/+RN/+Nm7aCij4YeLw3w8y8MOVMacf5sFZW19Y3Ngtbxe2d3b390sHhnUoySahPEp7ITogV5UxQXzPNaSeVFMchp+1wfJn77XsqFUvErZ6kNIjxULCIEayNdCP6br9Uduxa1au4F8iQWtWt1nNy7ngNB7m2M0cZlmj1S++9QUKymApNOFaq6zqpDqZYakY4nRV7maIpJmM8pF1DBY6pCqbzU2fo1CgDFCXSlNBorn6fmOJYqUkcms4Y65H67eXiX14301EjmDKRZpoKslgUZRzpBOV/owGTlGg+MQQTycytiIywxESbdIomhK9P0f/Er9ie7VzXyk1vmUYBjuEEzsCFOjThClrgA4EhPMATPFvcerRerNdF64q1nDmCH7DePgHQjo2p</latexit>

n2<latexit sha1_base64="gXHS5zmVWA/MH4ihOdYHRSLQaaw=">AAAB6XicdVBNS8NAEJ34WetX1aOXxSJ4CklbbXMrePFY0dhCG8pmu22XbjZhdyOU0J/gxYOKV/+RN/+Nm7aCij4YeLw3w8y8MOFMacf5sFZW19Y3Ngtbxe2d3b390sHhnYpTSahPYh7LTogV5UxQXzPNaSeRFEchp+1wcpn77XsqFYvFrZ4mNIjwSLAhI1gb6Ub0K/1S2bFrVa/iXiBDalW3Ws/JueM1HOTazhxlWKLVL733BjFJIyo04VipruskOsiw1IxwOiv2UkUTTCZ4RLuGChxRFWTzU2fo1CgDNIylKaHRXP0+keFIqWkUms4I67H67eXiX1431cNGkDGRpJoKslg0TDnSMcr/RgMmKdF8aggmkplbERljiYk26RRNCF+fov+JX7E927mulZveMo0CHMMJnIELdWjCFbTABwIjeIAneLa49Wi9WK+L1hVrOXMEP2C9fQLSEY2q</latexit><latexit sha1_base64="gXHS5zmVWA/MH4ihOdYHRSLQaaw=">AAAB6XicdVBNS8NAEJ34WetX1aOXxSJ4CklbbXMrePFY0dhCG8pmu22XbjZhdyOU0J/gxYOKV/+RN/+Nm7aCij4YeLw3w8y8MOFMacf5sFZW19Y3Ngtbxe2d3b390sHhnYpTSahPYh7LTogV5UxQXzPNaSeRFEchp+1wcpn77XsqFYvFrZ4mNIjwSLAhI1gb6Ub0K/1S2bFrVa/iXiBDalW3Ws/JueM1HOTazhxlWKLVL733BjFJIyo04VipruskOsiw1IxwOiv2UkUTTCZ4RLuGChxRFWTzU2fo1CgDNIylKaHRXP0+keFIqWkUms4I67H67eXiX1431cNGkDGRpJoKslg0TDnSMcr/RgMmKdF8aggmkplbERljiYk26RRNCF+fov+JX7E927mulZveMo0CHMMJnIELdWjCFbTABwIjeIAneLa49Wi9WK+L1hVrOXMEP2C9fQLSEY2q</latexit><latexit sha1_base64="gXHS5zmVWA/MH4ihOdYHRSLQaaw=">AAAB6XicdVBNS8NAEJ34WetX1aOXxSJ4CklbbXMrePFY0dhCG8pmu22XbjZhdyOU0J/gxYOKV/+RN/+Nm7aCij4YeLw3w8y8MOFMacf5sFZW19Y3Ngtbxe2d3b390sHhnYpTSahPYh7LTogV5UxQXzPNaSeRFEchp+1wcpn77XsqFYvFrZ4mNIjwSLAhI1gb6Ub0K/1S2bFrVa/iXiBDalW3Ws/JueM1HOTazhxlWKLVL733BjFJIyo04VipruskOsiw1IxwOiv2UkUTTCZ4RLuGChxRFWTzU2fo1CgDNIylKaHRXP0+keFIqWkUms4I67H67eXiX1431cNGkDGRpJoKslg0TDnSMcr/RgMmKdF8aggmkplbERljiYk26RRNCF+fov+JX7E927mulZveMo0CHMMJnIELdWjCFbTABwIjeIAneLa49Wi9WK+L1hVrOXMEP2C9fQLSEY2q</latexit>

ni<latexit sha1_base64="2TsETHPB09/fdJFNCwp4Zjlex/s=">AAAB6XicdVBNS8NAEJ34WetX1aOXxSJ4CklbbXMrePFY0dhCG8pmu2mXbjZhdyOU0p/gxYOKV/+RN/+Nm7aCij4YeLw3w8y8MOVMacf5sFZW19Y3Ngtbxe2d3b390sHhnUoySahPEp7ITogV5UxQXzPNaSeVFMchp+1wfJn77XsqFUvErZ6kNIjxULCIEayNdCP6rF8qO3at6lXcC2RIrepW6zk5d7yGg1zbmaMMS7T6pffeICFZTIUmHCvVdZ1UB1MsNSOczoq9TNEUkzEe0q6hAsdUBdP5qTN0apQBihJpSmg0V79PTHGs1CQOTWeM9Uj99nLxL6+b6agRTJlIM00FWSyKMo50gvK/0YBJSjSfGIKJZOZWREZYYqJNOkUTwten6H/iV2zPdq5r5aa3TKMAx3ACZ+BCHZpwBS3wgcAQHuAJni1uPVov1uuidcVazhzBD1hvnyVFjeE=</latexit><latexit sha1_base64="2TsETHPB09/fdJFNCwp4Zjlex/s=">AAAB6XicdVBNS8NAEJ34WetX1aOXxSJ4CklbbXMrePFY0dhCG8pmu2mXbjZhdyOU0p/gxYOKV/+RN/+Nm7aCij4YeLw3w8y8MOVMacf5sFZW19Y3Ngtbxe2d3b390sHhnUoySahPEp7ITogV5UxQXzPNaSeVFMchp+1wfJn77XsqFUvErZ6kNIjxULCIEayNdCP6rF8qO3at6lXcC2RIrepW6zk5d7yGg1zbmaMMS7T6pffeICFZTIUmHCvVdZ1UB1MsNSOczoq9TNEUkzEe0q6hAsdUBdP5qTN0apQBihJpSmg0V79PTHGs1CQOTWeM9Uj99nLxL6+b6agRTJlIM00FWSyKMo50gvK/0YBJSjSfGIKJZOZWREZYYqJNOkUTwten6H/iV2zPdq5r5aa3TKMAx3ACZ+BCHZpwBS3wgcAQHuAJni1uPVov1uuidcVazhzBD1hvnyVFjeE=</latexit><latexit sha1_base64="2TsETHPB09/fdJFNCwp4Zjlex/s=">AAAB6XicdVBNS8NAEJ34WetX1aOXxSJ4CklbbXMrePFY0dhCG8pmu2mXbjZhdyOU0p/gxYOKV/+RN/+Nm7aCij4YeLw3w8y8MOVMacf5sFZW19Y3Ngtbxe2d3b390sHhnUoySahPEp7ITogV5UxQXzPNaSeVFMchp+1wfJn77XsqFUvErZ6kNIjxULCIEayNdCP6rF8qO3at6lXcC2RIrepW6zk5d7yGg1zbmaMMS7T6pffeICFZTIUmHCvVdZ1UB1MsNSOczoq9TNEUkzEe0q6hAsdUBdP5qTN0apQBihJpSmg0V79PTHGs1CQOTWeM9Uj99nLxL6+b6agRTJlIM00FWSyKMo50gvK/0YBJSjSfGIKJZOZWREZYYqJNOkUTwten6H/iV2zPdq5r5aa3TKMAx3ACZ+BCHZpwBS3wgcAQHuAJni1uPVov1uuidcVazhzBD1hvnyVFjeE=</latexit>

La statistica di Maxwell—Boltzmann

g1<latexit sha1_base64="LQ3N1OCSNUNRezAzjJ8x0a6/ltY=">AAAB6XicdVBNS8NAEN34WetX1aOXxSJ4Cpu22uZW8OKxorGFNpTNdpMu3WzC7kYooT/BiwcVr/4jb/4bN20FFX0w8Hhvhpl5QcqZ0gh9WCura+sbm6Wt8vbO7t5+5eDwTiWZJNQjCU9kL8CKciaop5nmtJdKiuOA024wuSz87j2ViiXiVk9T6sc4EixkBGsj3URDZ1ipIrtRd2vOBTSkUXfqzYKcI7eFoGOjOapgic6w8j4YJSSLqdCEY6X6Dkq1n2OpGeF0Vh5kiqaYTHBE+4YKHFPl5/NTZ/DUKCMYJtKU0HCufp/IcazUNA5MZ4z1WP32CvEvr5/psOXnTKSZpoIsFoUZhzqBxd9wxCQlmk8NwUQycyskYywx0Sadsgnh61P4P/Fqtmuj60a17S7TKIFjcALOgAOaoA2uQAd4gIAIPIAn8Gxx69F6sV4XrSvWcuYI/ID19gnF642i</latexit><latexit sha1_base64="LQ3N1OCSNUNRezAzjJ8x0a6/ltY=">AAAB6XicdVBNS8NAEN34WetX1aOXxSJ4Cpu22uZW8OKxorGFNpTNdpMu3WzC7kYooT/BiwcVr/4jb/4bN20FFX0w8Hhvhpl5QcqZ0gh9WCura+sbm6Wt8vbO7t5+5eDwTiWZJNQjCU9kL8CKciaop5nmtJdKiuOA024wuSz87j2ViiXiVk9T6sc4EixkBGsj3URDZ1ipIrtRd2vOBTSkUXfqzYKcI7eFoGOjOapgic6w8j4YJSSLqdCEY6X6Dkq1n2OpGeF0Vh5kiqaYTHBE+4YKHFPl5/NTZ/DUKCMYJtKU0HCufp/IcazUNA5MZ4z1WP32CvEvr5/psOXnTKSZpoIsFoUZhzqBxd9wxCQlmk8NwUQycyskYywx0Sadsgnh61P4P/Fqtmuj60a17S7TKIFjcALOgAOaoA2uQAd4gIAIPIAn8Gxx69F6sV4XrSvWcuYI/ID19gnF642i</latexit><latexit sha1_base64="LQ3N1OCSNUNRezAzjJ8x0a6/ltY=">AAAB6XicdVBNS8NAEN34WetX1aOXxSJ4Cpu22uZW8OKxorGFNpTNdpMu3WzC7kYooT/BiwcVr/4jb/4bN20FFX0w8Hhvhpl5QcqZ0gh9WCura+sbm6Wt8vbO7t5+5eDwTiWZJNQjCU9kL8CKciaop5nmtJdKiuOA024wuSz87j2ViiXiVk9T6sc4EixkBGsj3URDZ1ipIrtRd2vOBTSkUXfqzYKcI7eFoGOjOapgic6w8j4YJSSLqdCEY6X6Dkq1n2OpGeF0Vh5kiqaYTHBE+4YKHFPl5/NTZ/DUKCMYJtKU0HCufp/IcazUNA5MZ4z1WP32CvEvr5/psOXnTKSZpoIsFoUZhzqBxd9wxCQlmk8NwUQycyskYywx0Sadsgnh61P4P/Fqtmuj60a17S7TKIFjcALOgAOaoA2uQAd4gIAIPIAn8Gxx69F6sV4XrSvWcuYI/ID19gnF642i</latexit>

g2<latexit sha1_base64="rZQjqZwAzGpZfhqyj33Vl7EVQg8=">AAAB6XicdVBNS8NAEN34WetX1aOXxSJ4Cpu22uZW8OKxorGFNpTNdpMu3WzC7kYooT/BiwcVr/4jb/4bN20FFX0w8Hhvhpl5QcqZ0gh9WCura+sbm6Wt8vbO7t5+5eDwTiWZJNQjCU9kL8CKciaop5nmtJdKiuOA024wuSz87j2ViiXiVk9T6sc4EixkBGsj3UTD2rBSRXaj7tacC2hIo+7UmwU5R24LQcdGc1TBEp1h5X0wSkgWU6EJx0r1HZRqP8dSM8LprDzIFE0xmeCI9g0VOKbKz+enzuCpUUYwTKQpoeFc/T6R41ipaRyYzhjrsfrtFeJfXj/TYcvPmUgzTQVZLAozDnUCi7/hiElKNJ8agolk5lZIxlhiok06ZRPC16fwf+LVbNdG141q212mUQLH4AScAQc0QRtcgQ7wAAEReABP4Nni1qP1Yr0uWles5cwR+AHr7RPHbo2j</latexit><latexit sha1_base64="rZQjqZwAzGpZfhqyj33Vl7EVQg8=">AAAB6XicdVBNS8NAEN34WetX1aOXxSJ4Cpu22uZW8OKxorGFNpTNdpMu3WzC7kYooT/BiwcVr/4jb/4bN20FFX0w8Hhvhpl5QcqZ0gh9WCura+sbm6Wt8vbO7t5+5eDwTiWZJNQjCU9kL8CKciaop5nmtJdKiuOA024wuSz87j2ViiXiVk9T6sc4EixkBGsj3UTD2rBSRXaj7tacC2hIo+7UmwU5R24LQcdGc1TBEp1h5X0wSkgWU6EJx0r1HZRqP8dSM8LprDzIFE0xmeCI9g0VOKbKz+enzuCpUUYwTKQpoeFc/T6R41ipaRyYzhjrsfrtFeJfXj/TYcvPmUgzTQVZLAozDnUCi7/hiElKNJ8agolk5lZIxlhiok06ZRPC16fwf+LVbNdG141q212mUQLH4AScAQc0QRtcgQ7wAAEReABP4Nni1qP1Yr0uWles5cwR+AHr7RPHbo2j</latexit><latexit sha1_base64="rZQjqZwAzGpZfhqyj33Vl7EVQg8=">AAAB6XicdVBNS8NAEN34WetX1aOXxSJ4Cpu22uZW8OKxorGFNpTNdpMu3WzC7kYooT/BiwcVr/4jb/4bN20FFX0w8Hhvhpl5QcqZ0gh9WCura+sbm6Wt8vbO7t5+5eDwTiWZJNQjCU9kL8CKciaop5nmtJdKiuOA024wuSz87j2ViiXiVk9T6sc4EixkBGsj3UTD2rBSRXaj7tacC2hIo+7UmwU5R24LQcdGc1TBEp1h5X0wSkgWU6EJx0r1HZRqP8dSM8LprDzIFE0xmeCI9g0VOKbKz+enzuCpUUYwTKQpoeFc/T6R41ipaRyYzhjrsfrtFeJfXj/TYcvPmUgzTQVZLAozDnUCi7/hiElKNJ8agolk5lZIxlhiok06ZRPC16fwf+LVbNdG141q212mUQLH4AScAQc0QRtcgQ7wAAEReABP4Nni1qP1Yr0uWles5cwR+AHr7RPHbo2j</latexit>

gi<latexit sha1_base64="NvsM2T/CqZFWR+IOB/1j+VGwRzI=">AAAB6XicdVBNS8NAEN34WetX1aOXxSJ4Cpu22uZW8OKxorGFNpTNdpMu3WzC7kYooT/BiwcVr/4jb/4bN20FFX0w8Hhvhpl5QcqZ0gh9WCura+sbm6Wt8vbO7t5+5eDwTiWZJNQjCU9kL8CKciaop5nmtJdKiuOA024wuSz87j2ViiXiVk9T6sc4EixkBGsj3URDNqxUkd2ouzXnAhrSqDv1ZkHOkdtC0LHRHFWwRGdYeR+MEpLFVGjCsVJ9B6Xaz7HUjHA6Kw8yRVNMJjiifUMFjqny8/mpM3hqlBEME2lKaDhXv0/kOFZqGgemM8Z6rH57hfiX18902PJzJtJMU0EWi8KMQ53A4m84YpISzaeGYCKZuRWSMZaYaJNO2YTw9Sn8n3g127XRdaPadpdplMAxOAFnwAFN0AZXoAM8QEAEHsATeLa49Wi9WK+L1hVrOXMEfsB6+wQaoo3a</latexit><latexit sha1_base64="NvsM2T/CqZFWR+IOB/1j+VGwRzI=">AAAB6XicdVBNS8NAEN34WetX1aOXxSJ4Cpu22uZW8OKxorGFNpTNdpMu3WzC7kYooT/BiwcVr/4jb/4bN20FFX0w8Hhvhpl5QcqZ0gh9WCura+sbm6Wt8vbO7t5+5eDwTiWZJNQjCU9kL8CKciaop5nmtJdKiuOA024wuSz87j2ViiXiVk9T6sc4EixkBGsj3URDNqxUkd2ouzXnAhrSqDv1ZkHOkdtC0LHRHFWwRGdYeR+MEpLFVGjCsVJ9B6Xaz7HUjHA6Kw8yRVNMJjiifUMFjqny8/mpM3hqlBEME2lKaDhXv0/kOFZqGgemM8Z6rH57hfiX18902PJzJtJMU0EWi8KMQ53A4m84YpISzaeGYCKZuRWSMZaYaJNO2YTw9Sn8n3g127XRdaPadpdplMAxOAFnwAFN0AZXoAM8QEAEHsATeLa49Wi9WK+L1hVrOXMEfsB6+wQaoo3a</latexit><latexit sha1_base64="NvsM2T/CqZFWR+IOB/1j+VGwRzI=">AAAB6XicdVBNS8NAEN34WetX1aOXxSJ4Cpu22uZW8OKxorGFNpTNdpMu3WzC7kYooT/BiwcVr/4jb/4bN20FFX0w8Hhvhpl5QcqZ0gh9WCura+sbm6Wt8vbO7t5+5eDwTiWZJNQjCU9kL8CKciaop5nmtJdKiuOA024wuSz87j2ViiXiVk9T6sc4EixkBGsj3URDNqxUkd2ouzXnAhrSqDv1ZkHOkdtC0LHRHFWwRGdYeR+MEpLFVGjCsVJ9B6Xaz7HUjHA6Kw8yRVNMJjiifUMFjqny8/mpM3hqlBEME2lKaDhXv0/kOFZqGgemM8Z6rH57hfiX18902PJzJtJMU0EWi8KMQ53A4m84YpISzaeGYCKZuRWSMZaYaJNO2YTw9Sn8n3g127XRdaPadpdplMAxOAFnwAFN0AZXoAM8QEAEHsATeLa49Wi9WK+L1hVrOXMEfsB6+wQaoo3a</latexit>

W = N !Y

i

gnii

ni!<latexit sha1_base64="q6yeXsuDR0vyNCcOluT6zF8yO2E=">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</latexit><latexit sha1_base64="q6yeXsuDR0vyNCcOluT6zF8yO2E=">AAACDXicdVDLTgIxFO34RHyNunRTJCSuyAygwEJD4saVwUSEBHDSKR1o6HQmbceETOYL3PgrblyocevenX9jBzBRoye5Nyfn3Jv2HjdkVCrL+jAWFpeWV1Yza9n1jc2tbXNn91oGkcCkhQMWiI6LJGGUk5aiipFOKAjyXUba7vgs9du3REga8Cs1CUnfR0NOPYqR0pJjFtrwBF7kYC8UwcChsOcJhOOhQ29i7tAkSXsuccy8VayU6yX7GGpSKdvlakqOrHrNgnbRmiIP5mg65ntvEODIJ1xhhqTs2lao+jESimJGkmwvkiREeIyGpKspRz6R/Xh6TgILWhlALxC6uIJT9ftGjHwpJ76rJ32kRvK3l4p/ed1IebV+THkYKcLx7CEvYlAFMM0GDqggWLGJJggLqv8K8QjpQJROMKtD+LoU/k9apWK9aF1W8o3TeRoZsA8OwCGwQRU0wDloghbA4A48gCfwbNwbj8aL8TobXTDmO3vgB4y3T7qOm4s=</latexit><latexit sha1_base64="q6yeXsuDR0vyNCcOluT6zF8yO2E=">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</latexit>

Se si assumono le particelle indistinguibili, nel limite (bassa densità) si ottiene:gi ≫ ni

W =Y

i

gnii

ni!<latexit sha1_base64="clZ+HGdKz65Or6RmK/SISqblUaE=">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</latexit><latexit sha1_base64="clZ+HGdKz65Or6RmK/SISqblUaE=">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</latexit><latexit sha1_base64="clZ+HGdKz65Or6RmK/SISqblUaE=">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</latexit>

Consideriamo un sistema isolato formato da un gran numero N di particelle.

Se il livello ha una degenerazione , le particelle possono occupare uno qualsiasi di questi livelli.

Ei gi ni

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La statistica di Maxwell—Boltzmann

g1<latexit sha1_base64="LQ3N1OCSNUNRezAzjJ8x0a6/ltY=">AAAB6XicdVBNS8NAEN34WetX1aOXxSJ4Cpu22uZW8OKxorGFNpTNdpMu3WzC7kYooT/BiwcVr/4jb/4bN20FFX0w8Hhvhpl5QcqZ0gh9WCura+sbm6Wt8vbO7t5+5eDwTiWZJNQjCU9kL8CKciaop5nmtJdKiuOA024wuSz87j2ViiXiVk9T6sc4EixkBGsj3URDZ1ipIrtRd2vOBTSkUXfqzYKcI7eFoGOjOapgic6w8j4YJSSLqdCEY6X6Dkq1n2OpGeF0Vh5kiqaYTHBE+4YKHFPl5/NTZ/DUKCMYJtKU0HCufp/IcazUNA5MZ4z1WP32CvEvr5/psOXnTKSZpoIsFoUZhzqBxd9wxCQlmk8NwUQycyskYywx0Sadsgnh61P4P/Fqtmuj60a17S7TKIFjcALOgAOaoA2uQAd4gIAIPIAn8Gxx69F6sV4XrSvWcuYI/ID19gnF642i</latexit><latexit sha1_base64="LQ3N1OCSNUNRezAzjJ8x0a6/ltY=">AAAB6XicdVBNS8NAEN34WetX1aOXxSJ4Cpu22uZW8OKxorGFNpTNdpMu3WzC7kYooT/BiwcVr/4jb/4bN20FFX0w8Hhvhpl5QcqZ0gh9WCura+sbm6Wt8vbO7t5+5eDwTiWZJNQjCU9kL8CKciaop5nmtJdKiuOA024wuSz87j2ViiXiVk9T6sc4EixkBGsj3URDZ1ipIrtRd2vOBTSkUXfqzYKcI7eFoGOjOapgic6w8j4YJSSLqdCEY6X6Dkq1n2OpGeF0Vh5kiqaYTHBE+4YKHFPl5/NTZ/DUKCMYJtKU0HCufp/IcazUNA5MZ4z1WP32CvEvr5/psOXnTKSZpoIsFoUZhzqBxd9wxCQlmk8NwUQycyskYywx0Sadsgnh61P4P/Fqtmuj60a17S7TKIFjcALOgAOaoA2uQAd4gIAIPIAn8Gxx69F6sV4XrSvWcuYI/ID19gnF642i</latexit><latexit sha1_base64="LQ3N1OCSNUNRezAzjJ8x0a6/ltY=">AAAB6XicdVBNS8NAEN34WetX1aOXxSJ4Cpu22uZW8OKxorGFNpTNdpMu3WzC7kYooT/BiwcVr/4jb/4bN20FFX0w8Hhvhpl5QcqZ0gh9WCura+sbm6Wt8vbO7t5+5eDwTiWZJNQjCU9kL8CKciaop5nmtJdKiuOA024wuSz87j2ViiXiVk9T6sc4EixkBGsj3URDZ1ipIrtRd2vOBTSkUXfqzYKcI7eFoGOjOapgic6w8j4YJSSLqdCEY6X6Dkq1n2OpGeF0Vh5kiqaYTHBE+4YKHFPl5/NTZ/DUKCMYJtKU0HCufp/IcazUNA5MZ4z1WP32CvEvr5/psOXnTKSZpoIsFoUZhzqBxd9wxCQlmk8NwUQycyskYywx0Sadsgnh61P4P/Fqtmuj60a17S7TKIFjcALOgAOaoA2uQAd4gIAIPIAn8Gxx69F6sV4XrSvWcuYI/ID19gnF642i</latexit>

g2<latexit sha1_base64="rZQjqZwAzGpZfhqyj33Vl7EVQg8=">AAAB6XicdVBNS8NAEN34WetX1aOXxSJ4Cpu22uZW8OKxorGFNpTNdpMu3WzC7kYooT/BiwcVr/4jb/4bN20FFX0w8Hhvhpl5QcqZ0gh9WCura+sbm6Wt8vbO7t5+5eDwTiWZJNQjCU9kL8CKciaop5nmtJdKiuOA024wuSz87j2ViiXiVk9T6sc4EixkBGsj3UTD2rBSRXaj7tacC2hIo+7UmwU5R24LQcdGc1TBEp1h5X0wSkgWU6EJx0r1HZRqP8dSM8LprDzIFE0xmeCI9g0VOKbKz+enzuCpUUYwTKQpoeFc/T6R41ipaRyYzhjrsfrtFeJfXj/TYcvPmUgzTQVZLAozDnUCi7/hiElKNJ8agolk5lZIxlhiok06ZRPC16fwf+LVbNdG141q212mUQLH4AScAQc0QRtcgQ7wAAEReABP4Nni1qP1Yr0uWles5cwR+AHr7RPHbo2j</latexit><latexit sha1_base64="rZQjqZwAzGpZfhqyj33Vl7EVQg8=">AAAB6XicdVBNS8NAEN34WetX1aOXxSJ4Cpu22uZW8OKxorGFNpTNdpMu3WzC7kYooT/BiwcVr/4jb/4bN20FFX0w8Hhvhpl5QcqZ0gh9WCura+sbm6Wt8vbO7t5+5eDwTiWZJNQjCU9kL8CKciaop5nmtJdKiuOA024wuSz87j2ViiXiVk9T6sc4EixkBGsj3UTD2rBSRXaj7tacC2hIo+7UmwU5R24LQcdGc1TBEp1h5X0wSkgWU6EJx0r1HZRqP8dSM8LprDzIFE0xmeCI9g0VOKbKz+enzuCpUUYwTKQpoeFc/T6R41ipaRyYzhjrsfrtFeJfXj/TYcvPmUgzTQVZLAozDnUCi7/hiElKNJ8agolk5lZIxlhiok06ZRPC16fwf+LVbNdG141q212mUQLH4AScAQc0QRtcgQ7wAAEReABP4Nni1qP1Yr0uWles5cwR+AHr7RPHbo2j</latexit><latexit sha1_base64="rZQjqZwAzGpZfhqyj33Vl7EVQg8=">AAAB6XicdVBNS8NAEN34WetX1aOXxSJ4Cpu22uZW8OKxorGFNpTNdpMu3WzC7kYooT/BiwcVr/4jb/4bN20FFX0w8Hhvhpl5QcqZ0gh9WCura+sbm6Wt8vbO7t5+5eDwTiWZJNQjCU9kL8CKciaop5nmtJdKiuOA024wuSz87j2ViiXiVk9T6sc4EixkBGsj3UTD2rBSRXaj7tacC2hIo+7UmwU5R24LQcdGc1TBEp1h5X0wSkgWU6EJx0r1HZRqP8dSM8LprDzIFE0xmeCI9g0VOKbKz+enzuCpUUYwTKQpoeFc/T6R41ipaRyYzhjrsfrtFeJfXj/TYcvPmUgzTQVZLAozDnUCi7/hiElKNJ8agolk5lZIxlhiok06ZRPC16fwf+LVbNdG141q212mUQLH4AScAQc0QRtcgQ7wAAEReABP4Nni1qP1Yr0uWles5cwR+AHr7RPHbo2j</latexit>

gi<latexit sha1_base64="NvsM2T/CqZFWR+IOB/1j+VGwRzI=">AAAB6XicdVBNS8NAEN34WetX1aOXxSJ4Cpu22uZW8OKxorGFNpTNdpMu3WzC7kYooT/BiwcVr/4jb/4bN20FFX0w8Hhvhpl5QcqZ0gh9WCura+sbm6Wt8vbO7t5+5eDwTiWZJNQjCU9kL8CKciaop5nmtJdKiuOA024wuSz87j2ViiXiVk9T6sc4EixkBGsj3URDNqxUkd2ouzXnAhrSqDv1ZkHOkdtC0LHRHFWwRGdYeR+MEpLFVGjCsVJ9B6Xaz7HUjHA6Kw8yRVNMJjiifUMFjqny8/mpM3hqlBEME2lKaDhXv0/kOFZqGgemM8Z6rH57hfiX18902PJzJtJMU0EWi8KMQ53A4m84YpISzaeGYCKZuRWSMZaYaJNO2YTw9Sn8n3g127XRdaPadpdplMAxOAFnwAFN0AZXoAM8QEAEHsATeLa49Wi9WK+L1hVrOXMEfsB6+wQaoo3a</latexit><latexit sha1_base64="NvsM2T/CqZFWR+IOB/1j+VGwRzI=">AAAB6XicdVBNS8NAEN34WetX1aOXxSJ4Cpu22uZW8OKxorGFNpTNdpMu3WzC7kYooT/BiwcVr/4jb/4bN20FFX0w8Hhvhpl5QcqZ0gh9WCura+sbm6Wt8vbO7t5+5eDwTiWZJNQjCU9kL8CKciaop5nmtJdKiuOA024wuSz87j2ViiXiVk9T6sc4EixkBGsj3URDNqxUkd2ouzXnAhrSqDv1ZkHOkdtC0LHRHFWwRGdYeR+MEpLFVGjCsVJ9B6Xaz7HUjHA6Kw8yRVNMJjiifUMFjqny8/mpM3hqlBEME2lKaDhXv0/kOFZqGgemM8Z6rH57hfiX18902PJzJtJMU0EWi8KMQ53A4m84YpISzaeGYCKZuRWSMZaYaJNO2YTw9Sn8n3g127XRdaPadpdplMAxOAFnwAFN0AZXoAM8QEAEHsATeLa49Wi9WK+L1hVrOXMEfsB6+wQaoo3a</latexit><latexit sha1_base64="NvsM2T/CqZFWR+IOB/1j+VGwRzI=">AAAB6XicdVBNS8NAEN34WetX1aOXxSJ4Cpu22uZW8OKxorGFNpTNdpMu3WzC7kYooT/BiwcVr/4jb/4bN20FFX0w8Hhvhpl5QcqZ0gh9WCura+sbm6Wt8vbO7t5+5eDwTiWZJNQjCU9kL8CKciaop5nmtJdKiuOA024wuSz87j2ViiXiVk9T6sc4EixkBGsj3URDNqxUkd2ouzXnAhrSqDv1ZkHOkdtC0LHRHFWwRGdYeR+MEpLFVGjCsVJ9B6Xaz7HUjHA6Kw8yRVNMJjiifUMFjqny8/mpM3hqlBEME2lKaDhXv0/kOFZqGgemM8Z6rH57hfiX18902PJzJtJMU0EWi8KMQ53A4m84YpISzaeGYCKZuRWSMZaYaJNO2YTw9Sn8n3g127XRdaPadpdplMAxOAFnwAFN0AZXoAM8QEAEHsATeLa49Wi9WK+L1hVrOXMEfsB6+wQaoo3a</latexit>

Calcoliamo ora la partizione più probabile con i vincoli del numero di particelle e dell’energia totale che devono rimangono costanti (sistema chiuso e isolato):

N =X

i

ni

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U =X

i

niEi

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Se si assumono le particelle indistinguibili, nel limite (bassa densità) si ottiene:gi ≫ ni

W =Y

i

gnii

ni!<latexit sha1_base64="clZ+HGdKz65Or6RmK/SISqblUaE=">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</latexit><latexit sha1_base64="clZ+HGdKz65Or6RmK/SISqblUaE=">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</latexit><latexit sha1_base64="clZ+HGdKz65Or6RmK/SISqblUaE=">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</latexit>

Consideriamo un sistema isolato formato da un gran numero N di particelle.

Per comodità consideriamo il logaritmo di W:

La statistica di Maxwell—Boltzmann

e troviamo il massimo di rispetto alla partizione con il metodo dei moltiplicatori di Lagrange:

ln W {n1, n2, …}

@

@ni

(X

i

[ni ln gi � ln(ni!)] + ↵

N �

X

i

ni

!+ �

U �

X

i

niEi

!)= 0

<latexit sha1_base64="Awxe/wl7AG5NJLpILT/vVfORNvY=">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</latexit><latexit sha1_base64="Awxe/wl7AG5NJLpILT/vVfORNvY=">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</latexit><latexit sha1_base64="Awxe/wl7AG5NJLpILT/vVfORNvY=">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</latexit>

Se possiamo usare la formula di Stirling , quindi:ni ≫ 1 ln(ni!) ≈ ni ln ni − ni

@

@ni

(X

i

[ni ln gi � ni lnni + ni] + ↵

N �

X

i

ni

!+ �

U �

X

i

niEi

!)= 0

<latexit sha1_base64="j6fLSBPuHVKhAVE8gUFI+tALeXk=">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</latexit><latexit sha1_base64="j6fLSBPuHVKhAVE8gUFI+tALeXk=">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</latexit><latexit sha1_base64="j6fLSBPuHVKhAVE8gUFI+tALeXk=">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</latexit>

W =Y

i

gnii

ni!) lnW =

X

i

[ni ln gi � ln(ni!)]<latexit sha1_base64="n3N4QoXEb20Wr/LeFBnaxZ1pGts=">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</latexit><latexit sha1_base64="n3N4QoXEb20Wr/LeFBnaxZ1pGts=">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</latexit><latexit sha1_base64="n3N4QoXEb20Wr/LeFBnaxZ1pGts=">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</latexit>

Otteniamo:

La statistica di Maxwell—Boltzmann

) ni = gie�↵��Ei

<latexit sha1_base64="6B5wWIjDT+nrTbl5liu7nIko7sQ=">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</latexit><latexit sha1_base64="6B5wWIjDT+nrTbl5liu7nIko7sQ=">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</latexit><latexit sha1_base64="6B5wWIjDT+nrTbl5liu7nIko7sQ=">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</latexit>

Dalla condizione sul numero di particelle otteniamo:

X

i

ni = N ) e�↵ =N

Z<latexit sha1_base64="rYd7DVRtKfXCuEK4BTJtLBnnrGk=">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</latexit><latexit sha1_base64="rYd7DVRtKfXCuEK4BTJtLBnnrGk=">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</latexit><latexit sha1_base64="rYd7DVRtKfXCuEK4BTJtLBnnrGk=">AAACKHicdVDLSgMxFM3Ud31VXboJFsGNZaatjy6EghtXomJV7NRyJ820oZnMmGSUMszvuPFX3CiouPVLzNQKKnrgwuGcc0nu8SLOlLbtNys3Nj4xOTU9k5+dm19YLCwtn6kwloQ2SMhDeeGBopwJ2tBMc3oRSQqBx+m519/P/PMbKhULxakeRLQVQFcwnxHQRmoX6q6KgzbDwswePsTudQwd7J6wbk+DlOHtSKFXyaYLPOpBanKuL4Ekh2lymbYLRbtUrdTKzjY2pFpxKjsZ2bJruzZ2SvYQRTTCUbvw5HZCEgdUaMJBqaZjR7qVgNSMcJrm3VjRCEgfurRpqICAqlYyvDTF60bpYD+UZoTGQ/X7RgKBUoPAM8kAdE/99jLxL68Za3+3lTARxZoK8vmQH3OsQ5zVhjtMUqL5wBAgkpm/YtID04I25eZNCV+X4v9Jo1yqlezjarFeG7UxjVbRGtpADtpBdXSAjlADEXSHHtAzerHurUfr1Xr7jOas0c4K+gHr/QM3HKZC</latexit>

Z =X

i

gie��Ei

<latexit sha1_base64="F4T/cccsMhrGez9DYW3hs1I9FWY=">AAACBXicdVDLSgNBEJyNrxhfUY8iDAbBi2E3iSZ7EAIieIzgmmASl9lJJxky+2BmVghLTl78FS8eVLz6D978GycPQUULGoqqbrq7vIgzqUzzw0jNzS8sLqWXMyura+sb2c2tKxnGgoJDQx6KhkckcBaAo5ji0IgEEN/jUPcGp2O/fgtCsjC4VMMI2j7pBazLKFFacrO71/gEt2Tsuwz3dMFNctjyQBF85rKRm82Z+VLRLljHWJNS0SqWx+TItCsmtvLmBDk0Q83Nvrc6IY19CBTlRMqmZUaqnRChGOUwyrRiCRGhA9KDpqYB8UG2k8kbI7yvlQ7uhkJXoPBE/T6REF/Koe/pTp+ovvztjcW/vGasupV2woIoVhDQ6aJuzLEK8TgT3GECqOJDTQgVTN+KaZ8IQpVOLqND+PoU/0+cQt7OmxelXNWepZFGO2gPHSALlVEVnaMachBFd+gBPaFn4954NF6M12lrypjNbKMfMN4+Ad5sl6w=</latexit><latexit sha1_base64="F4T/cccsMhrGez9DYW3hs1I9FWY=">AAACBXicdVDLSgNBEJyNrxhfUY8iDAbBi2E3iSZ7EAIieIzgmmASl9lJJxky+2BmVghLTl78FS8eVLz6D978GycPQUULGoqqbrq7vIgzqUzzw0jNzS8sLqWXMyura+sb2c2tKxnGgoJDQx6KhkckcBaAo5ji0IgEEN/jUPcGp2O/fgtCsjC4VMMI2j7pBazLKFFacrO71/gEt2Tsuwz3dMFNctjyQBF85rKRm82Z+VLRLljHWJNS0SqWx+TItCsmtvLmBDk0Q83Nvrc6IY19CBTlRMqmZUaqnRChGOUwyrRiCRGhA9KDpqYB8UG2k8kbI7yvlQ7uhkJXoPBE/T6REF/Koe/pTp+ovvztjcW/vGasupV2woIoVhDQ6aJuzLEK8TgT3GECqOJDTQgVTN+KaZ8IQpVOLqND+PoU/0+cQt7OmxelXNWepZFGO2gPHSALlVEVnaMachBFd+gBPaFn4954NF6M12lrypjNbKMfMN4+Ad5sl6w=</latexit><latexit sha1_base64="F4T/cccsMhrGez9DYW3hs1I9FWY=">AAACBXicdVDLSgNBEJyNrxhfUY8iDAbBi2E3iSZ7EAIieIzgmmASl9lJJxky+2BmVghLTl78FS8eVLz6D978GycPQUULGoqqbrq7vIgzqUzzw0jNzS8sLqWXMyura+sb2c2tKxnGgoJDQx6KhkckcBaAo5ji0IgEEN/jUPcGp2O/fgtCsjC4VMMI2j7pBazLKFFacrO71/gEt2Tsuwz3dMFNctjyQBF85rKRm82Z+VLRLljHWJNS0SqWx+TItCsmtvLmBDk0Q83Nvrc6IY19CBTlRMqmZUaqnRChGOUwyrRiCRGhA9KDpqYB8UG2k8kbI7yvlQ7uhkJXoPBE/T6REF/Koe/pTp+ovvztjcW/vGasupV2woIoVhDQ6aJuzLEK8TgT3GECqOJDTQgVTN+KaZ8IQpVOLqND+PoU/0+cQt7OmxelXNWepZFGO2gPHSALlVEVnaMachBFd+gBPaFn4954NF6M12lrypjNbKMfMN4+Ad5sl6w=</latexit>

funzione di partizione

Infine:

ni =N

Zgie

��Ei

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statistica di Maxwell—Boltzmann

ln gi � lnni � ↵� �Ei = 0<latexit sha1_base64="VU+Q+4TD+7jdwb6oumOElRqhRZo=">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</latexit><latexit sha1_base64="VU+Q+4TD+7jdwb6oumOElRqhRZo=">AAACEnicdZBNSwMxEIazflu/qh69DBZBBEvWVm0PgiCCRwWrQlvKbJq2odnskmSFUvofvPhXvHhQ8erJm//GbFtBRV8IPLwzw2TeIJbCWEo/vInJqemZ2bn5zMLi0vJKdnXtykSJZrzCIhnpmwANl0LxihVW8ptYcwwDya+D7klav77l2ohIXdpezOshtpVoCYbWWY3sTk0qaDcE7EJKakQo4w7Cbi3gFuHUeUdAoZHN0XyxUN7zD8BBseAXDlPYp+USBT9Ph8qRsc4b2fdaM2JJyJVlEo2p+jS29T5qK5jkg0wtMTxG1sU2rzpUGHJT7w9vGsCWc5rQirR7ysLQ/T7Rx9CYXhi4zhBtx/yupeZftWpiW6V6X6g4sVyx0aJWIsFGkAYETaE5s7LnAJkW7q/AOqiRWRdjxoXwdSn8D5W9fDlPL4q54/I4jTmyQTbJNvHJITkmZ+ScVAgjd+SBPJFn79579F6811HrhDeeWSc/5L19AnXMmu4=</latexit><latexit sha1_base64="VU+Q+4TD+7jdwb6oumOElRqhRZo=">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</latexit>

La statistica di Maxwell—Boltzmann: energia

ni =N

Zgie

��Ei

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Il valore di è legato all’energia del sistema:β

U =X

i

niEi =N

Z

X

i

giEie��Ei

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= �N

Z

d

d�

X

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| {z }Z

<latexit sha1_base64="xU35K+pQ59FueyWFBPsZVvVc4D8=">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</latexit><latexit sha1_base64="xU35K+pQ59FueyWFBPsZVvVc4D8=">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</latexit><latexit sha1_base64="xU35K+pQ59FueyWFBPsZVvVc4D8=">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</latexit>

U = �Nd

d�lnZ

<latexit sha1_base64="xxGcWw0uS/CaQVXqvcOF/X5jqNI=">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</latexit><latexit sha1_base64="xxGcWw0uS/CaQVXqvcOF/X5jqNI=">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</latexit><latexit sha1_base64="xxGcWw0uS/CaQVXqvcOF/X5jqNI=">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</latexit>

hEi = � d

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energia media per particella

La statistica di Maxwell—Boltzmann: temperatura

Fissato il sistema, il rapporto tra le occupazioni di due livelli dipende

solo da :

Ei > Ejβ

Per bassi valori di il rapporto cresce: si popolano i livelli a energia più alta.β

Per alti valori di il rapporto diminuisce: si popolano i livelli a energia più bassa.β

Possiamo intuire che esiste un legame di proporzionalità inversa tra tra e la temperatura assoluta T.

β

“Caldo”: bassi valori di e alta T.β

“Freddo”: alti valori di e bassa T.β

ni

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0 5 10 15 20

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� = 0.1<latexit sha1_base64="PtYC0aOl0tu2PyvdjpRuHdsTDVU=">AAAB8XicdVDLSgMxFM34rPVVdekmWARXQ6YdbbsQCm5cVnBsYTqUTJq2oZkHyR2hlH6GGxcqbv0bd/6NmbaCih4InJxzL/feE6ZSaCDkw1pZXVvf2CxsFbd3dvf2SweHdzrJFOMeS2SiOiHVXIqYeyBA8k6qOI1Cydvh+Cr32/dcaZHEtzBJeRDRYSwGglEwkt8NOVB8iYnt9EplYrvVRsW5MF/XrTrVWk7OSaNOsGOTOcpoiVav9N7tJyyLeAxMUq19h6QQTKkCwSSfFbuZ5illYzrkvqExjbgOpvOVZ/jUKH08SJR5MeC5+r1jSiOtJ1FoKiMKI/3by8W/PD+DQT2YijjNgMdsMWiQSQwJzu/HfaE4AzkxhDIlzK6YjaiiDExKRRPC16X4f+JV7IZNbtxy012mUUDH6ASdIQfVUBNdoxbyEEMJekBP6NkC69F6sV4XpSvWsucI/YD19gkWxY/6</latexit><latexit sha1_base64="PtYC0aOl0tu2PyvdjpRuHdsTDVU=">AAAB8XicdVDLSgMxFM34rPVVdekmWARXQ6YdbbsQCm5cVnBsYTqUTJq2oZkHyR2hlH6GGxcqbv0bd/6NmbaCih4InJxzL/feE6ZSaCDkw1pZXVvf2CxsFbd3dvf2SweHdzrJFOMeS2SiOiHVXIqYeyBA8k6qOI1Cydvh+Cr32/dcaZHEtzBJeRDRYSwGglEwkt8NOVB8iYnt9EplYrvVRsW5MF/XrTrVWk7OSaNOsGOTOcpoiVav9N7tJyyLeAxMUq19h6QQTKkCwSSfFbuZ5illYzrkvqExjbgOpvOVZ/jUKH08SJR5MeC5+r1jSiOtJ1FoKiMKI/3by8W/PD+DQT2YijjNgMdsMWiQSQwJzu/HfaE4AzkxhDIlzK6YjaiiDExKRRPC16X4f+JV7IZNbtxy012mUUDH6ASdIQfVUBNdoxbyEEMJekBP6NkC69F6sV4XpSvWsucI/YD19gkWxY/6</latexit><latexit sha1_base64="PtYC0aOl0tu2PyvdjpRuHdsTDVU=">AAAB8XicdVDLSgMxFM34rPVVdekmWARXQ6YdbbsQCm5cVnBsYTqUTJq2oZkHyR2hlH6GGxcqbv0bd/6NmbaCih4InJxzL/feE6ZSaCDkw1pZXVvf2CxsFbd3dvf2SweHdzrJFOMeS2SiOiHVXIqYeyBA8k6qOI1Cydvh+Cr32/dcaZHEtzBJeRDRYSwGglEwkt8NOVB8iYnt9EplYrvVRsW5MF/XrTrVWk7OSaNOsGOTOcpoiVav9N7tJyyLeAxMUq19h6QQTKkCwSSfFbuZ5illYzrkvqExjbgOpvOVZ/jUKH08SJR5MeC5+r1jSiOtJ1FoKiMKI/3by8W/PD+DQT2YijjNgMdsMWiQSQwJzu/HfaE4AzkxhDIlzK6YjaiiDExKRRPC16X4f+JV7IZNbtxy012mUUDH6ASdIQfVUBNdoxbyEEMJekBP6NkC69F6sV4XpSvWsucI/YD19gkWxY/6</latexit>

� = 0.2<latexit sha1_base64="x3MZuGyZ3vHnjGjH7oVQrjqWiAA=">AAAB8XicdVDLSgMxFM34rPVVdekmWARXQ6YdbbsQCm5cVrC2MB1KJs20oZkHyR2hDP0MNy5U3Po37vwb04egogcCJ+fcy733BKkUGgj5sFZW19Y3Ngtbxe2d3b390sHhnU4yxXibJTJR3YBqLkXM2yBA8m6qOI0CyTvB+Grmd+650iKJb2GScj+iw1iEglEwktcLOFB8iYld6ZfKxHarjYpzYb6uW3WqtRk5J406wY5N5iijJVr90ntvkLAs4jEwSbX2HJKCn1MFgkk+LfYyzVPKxnTIPUNjGnHt5/OVp/jUKAMcJsq8GPBc/d6R00jrSRSYyojCSP/2ZuJfnpdBWPdzEacZ8JgtBoWZxJDg2f14IBRnICeGUKaE2RWzEVWUgUmpaEL4uhT/T9oVu2GTG7fcdJdpFNAxOkFnyEE11ETXqIXaiKEEPaAn9GyB9Wi9WK+L0hVr2XOEfsB6+wQYSI/7</latexit><latexit sha1_base64="x3MZuGyZ3vHnjGjH7oVQrjqWiAA=">AAAB8XicdVDLSgMxFM34rPVVdekmWARXQ6YdbbsQCm5cVrC2MB1KJs20oZkHyR2hDP0MNy5U3Po37vwb04egogcCJ+fcy733BKkUGgj5sFZW19Y3Ngtbxe2d3b390sHhnU4yxXibJTJR3YBqLkXM2yBA8m6qOI0CyTvB+Grmd+650iKJb2GScj+iw1iEglEwktcLOFB8iYld6ZfKxHarjYpzYb6uW3WqtRk5J406wY5N5iijJVr90ntvkLAs4jEwSbX2HJKCn1MFgkk+LfYyzVPKxnTIPUNjGnHt5/OVp/jUKAMcJsq8GPBc/d6R00jrSRSYyojCSP/2ZuJfnpdBWPdzEacZ8JgtBoWZxJDg2f14IBRnICeGUKaE2RWzEVWUgUmpaEL4uhT/T9oVu2GTG7fcdJdpFNAxOkFnyEE11ETXqIXaiKEEPaAn9GyB9Wi9WK+L0hVr2XOEfsB6+wQYSI/7</latexit><latexit sha1_base64="x3MZuGyZ3vHnjGjH7oVQrjqWiAA=">AAAB8XicdVDLSgMxFM34rPVVdekmWARXQ6YdbbsQCm5cVrC2MB1KJs20oZkHyR2hDP0MNy5U3Po37vwb04egogcCJ+fcy733BKkUGgj5sFZW19Y3Ngtbxe2d3b390sHhnU4yxXibJTJR3YBqLkXM2yBA8m6qOI0CyTvB+Grmd+650iKJb2GScj+iw1iEglEwktcLOFB8iYld6ZfKxHarjYpzYb6uW3WqtRk5J406wY5N5iijJVr90ntvkLAs4jEwSbX2HJKCn1MFgkk+LfYyzVPKxnTIPUNjGnHt5/OVp/jUKAMcJsq8GPBc/d6R00jrSRSYyojCSP/2ZuJfnpdBWPdzEacZ8JgtBoWZxJDg2f14IBRnICeGUKaE2RWzEVWUgUmpaEL4uhT/T9oVu2GTG7fcdJdpFNAxOkFnyEE11ETXqIXaiKEEPaAn9GyB9Wi9WK+L0hVr2XOEfsB6+wQYSI/7</latexit>

� = 1.0<latexit sha1_base64="fWrE+72QwKTU2hOLlj7tOHIgT2I=">AAAB8XicdVDLSsNAFJ34rPVVdelmsAiuQtJG2y6EghuXFYwtpKFMppN26OTBzI1QQj/DjQsVt/6NO//GSVtBRQ8MHM45l7n3BKngCizrw1hZXVvf2Cxtlbd3dvf2KweHdyrJJGUuTUQiewFRTPCYucBBsF4qGYkCwbrB5Krwu/dMKp7EtzBNmR+RUcxDTgloyesHDAi+xLZpDSpVy3TqrZp9gTVx6na9UZBzq9W0isAcVbREZ1B57w8TmkUsBiqIUp5tpeDnRAKngs3K/UyxlNAJGTFP05hETPn5fOUZPtXKEIeJ1C8GPFe/T+QkUmoaBToZERir314h/uV5GYRNP+dxmgGL6eKjMBMYElzcj4dcMgpiqgmhkutdMR0TSSjolsq6hK9L8f/ErZkt07pxqm1n2UYJHaMTdIZs1EBtdI06yEUUJegBPaFnA4xH48V4XURXjOXMEfoB4+0TFseP+g==</latexit><latexit sha1_base64="fWrE+72QwKTU2hOLlj7tOHIgT2I=">AAAB8XicdVDLSsNAFJ34rPVVdelmsAiuQtJG2y6EghuXFYwtpKFMppN26OTBzI1QQj/DjQsVt/6NO//GSVtBRQ8MHM45l7n3BKngCizrw1hZXVvf2Cxtlbd3dvf2KweHdyrJJGUuTUQiewFRTPCYucBBsF4qGYkCwbrB5Krwu/dMKp7EtzBNmR+RUcxDTgloyesHDAi+xLZpDSpVy3TqrZp9gTVx6na9UZBzq9W0isAcVbREZ1B57w8TmkUsBiqIUp5tpeDnRAKngs3K/UyxlNAJGTFP05hETPn5fOUZPtXKEIeJ1C8GPFe/T+QkUmoaBToZERir314h/uV5GYRNP+dxmgGL6eKjMBMYElzcj4dcMgpiqgmhkutdMR0TSSjolsq6hK9L8f/ErZkt07pxqm1n2UYJHaMTdIZs1EBtdI06yEUUJegBPaFnA4xH48V4XURXjOXMEfoB4+0TFseP+g==</latexit><latexit sha1_base64="fWrE+72QwKTU2hOLlj7tOHIgT2I=">AAAB8XicdVDLSsNAFJ34rPVVdelmsAiuQtJG2y6EghuXFYwtpKFMppN26OTBzI1QQj/DjQsVt/6NO//GSVtBRQ8MHM45l7n3BKngCizrw1hZXVvf2Cxtlbd3dvf2KweHdyrJJGUuTUQiewFRTPCYucBBsF4qGYkCwbrB5Krwu/dMKp7EtzBNmR+RUcxDTgloyesHDAi+xLZpDSpVy3TqrZp9gTVx6na9UZBzq9W0isAcVbREZ1B57w8TmkUsBiqIUp5tpeDnRAKngs3K/UyxlNAJGTFP05hETPn5fOUZPtXKEIeJ1C8GPFe/T+QkUmoaBToZERir314h/uV5GYRNP+dxmgGL6eKjMBMYElzcj4dcMgpiqgmhkutdMR0TSSjolsq6hK9L8f/ErZkt07pxqm1n2UYJHaMTdIZs1EBtdI06yEUUJegBPaFnA4xH48V4XURXjOXMEfoB4+0TFseP+g==</latexit>

Equilibrio e massima entropia

Nel trovare la partizione che massimizza W abbiamo implicitamente utilizzato il cosiddetto principio di massima entropia.

Infatti, sfruttando la monotonia della funzione logaritmo, abbiamo cercato il massimo di . con il metodo dei moltiplicatori di Lagrange, per tenere conto dei vincoli sull’energia e il numero di particelle.

ln W

D’altra parte, secondo la formula di Boltzmann, l’entropia di un sistema è:

S = kB lnW<latexit sha1_base64="X5+5E7fm01LWfjhn1aPh7tQ8Dc0=">AAAB83icdVBNTwIxEO3iF+IX6tFLIzHxRLqAAgcN0YtHjK6QwIZ0Sxcaut217ZKQDb/Diwc1Xv0z3vw3dgETNfqSSV7em8nMPC/iTGmEPqzM0vLK6lp2PbexubW9k9/du1NhLAl1SMhD2fawopwJ6mimOW1HkuLA47TljS5TvzWmUrFQ3OpJRN0ADwTzGcHaSO4NPIOj3gXscgFbvXwBFSvlesk+hYZUyna5mpITVK8haBfRDAWwQLOXf+/2QxIHVGjCsVIdG0XaTbDUjHA6zXVjRSNMRnhAO4YKHFDlJrOjp/DIKH3oh9KU0HCmfp9IcKDUJPBMZ4D1UP32UvEvrxNrv+YmTESxpoLMF/kxhzqEaQKwzyQlmk8MwUQycyskQywx0SannAnh61P4P3FKxXoRXVcKjfNFGllwAA7BMbBBFTTAFWgCBxBwDx7AE3i2xtaj9WK9zlsz1mJmH/yA9fYJX6KQvQ==</latexit><latexit sha1_base64="X5+5E7fm01LWfjhn1aPh7tQ8Dc0=">AAAB83icdVBNTwIxEO3iF+IX6tFLIzHxRLqAAgcN0YtHjK6QwIZ0Sxcaut217ZKQDb/Diwc1Xv0z3vw3dgETNfqSSV7em8nMPC/iTGmEPqzM0vLK6lp2PbexubW9k9/du1NhLAl1SMhD2fawopwJ6mimOW1HkuLA47TljS5TvzWmUrFQ3OpJRN0ADwTzGcHaSO4NPIOj3gXscgFbvXwBFSvlesk+hYZUyna5mpITVK8haBfRDAWwQLOXf+/2QxIHVGjCsVIdG0XaTbDUjHA6zXVjRSNMRnhAO4YKHFDlJrOjp/DIKH3oh9KU0HCmfp9IcKDUJPBMZ4D1UP32UvEvrxNrv+YmTESxpoLMF/kxhzqEaQKwzyQlmk8MwUQycyskQywx0SannAnh61P4P3FKxXoRXVcKjfNFGllwAA7BMbBBFTTAFWgCBxBwDx7AE3i2xtaj9WK9zlsz1mJmH/yA9fYJX6KQvQ==</latexit><latexit sha1_base64="X5+5E7fm01LWfjhn1aPh7tQ8Dc0=">AAAB83icdVBNTwIxEO3iF+IX6tFLIzHxRLqAAgcN0YtHjK6QwIZ0Sxcaut217ZKQDb/Diwc1Xv0z3vw3dgETNfqSSV7em8nMPC/iTGmEPqzM0vLK6lp2PbexubW9k9/du1NhLAl1SMhD2fawopwJ6mimOW1HkuLA47TljS5TvzWmUrFQ3OpJRN0ADwTzGcHaSO4NPIOj3gXscgFbvXwBFSvlesk+hYZUyna5mpITVK8haBfRDAWwQLOXf+/2QxIHVGjCsVIdG0XaTbDUjHA6zXVjRSNMRnhAO4YKHFDlJrOjp/DIKH3oh9KU0HCmfp9IcKDUJPBMZ4D1UP32UvEvrxNrv+YmTESxpoLMF/kxhzqEaQKwzyQlmk8MwUQycyskQywx0SannAnh61P4P3FKxXoRXVcKjfNFGllwAA7BMbBBFTTAFWgCBxBwDx7AE3i2xtaj9WK9zlsz1mJmH/yA9fYJX6KQvQ==</latexit>

quindi la partizione più probabile che dà lo stato di equilibrio è anche quella che massimizza l’entropia, come, d’altra parte, ci si poteva attendere dalle considerazione sul secondo principio della termodinamica.

Gas perfetto, Maxwell—Boltzmann e temperatura

Per trovare il legame tra e T, consideriamo un gas perfetto monoatomico, dove i livelli energetici variano in modo continuo:

β

Osserviamo che l’energia dipende solo dal modulo p della quantità di moto: tutti i livelli che stanno sulla superficie di una sfera di raggio p sono degeneri.

Essendo in lo spazio delle fasi continuo, il numero di livelli degeneri è infinito…

E =1

2mv2 =

p2

2m<latexit sha1_base64="0zBSshwOGS6cpjhauSyk5dYAMv4=">AAACCnicdVDLSgMxFM34rPU16tJNaBFclZlpte1CKIjgsoK1hb7IpJk2NJkZkkyhDLN346+4caHi1i9w59+Y6QNU9MCFk3PuJfceN2RUKsv6NFZW19Y3NjNb2e2d3b198+DwTgaRwKSBAxaIloskYdQnDUUVI61QEMRdRpru+DL1mxMiJA38WzUNSZejoU89ipHSUt/MXcEL2PEEwrYDOZz0nOU7DntOEjs86Zt5q1AqVh37HGpSKtrFckrOrGrFgnbBmiEPFqj3zY/OIMARJ77CDEnZtq1QdWMkFMWMJNlOJEmI8BgNSVtTH3Eiu/HslgSeaGUAvUDo8hWcqd8nYsSlnHJXd3KkRvK3l4p/ee1IeZVuTP0wUsTH84+8iEEVwDQYOKCCYMWmmiAsqN4V4hHSQSgdX1aHsLwU/k8aTqFasG5K+Vp1kUYGHIMcOAU2KIMauAZ10AAY3INH8AxejAfjyXg13uatK8Zi5gj8gPH+BUbqmOY=</latexit><latexit sha1_base64="0zBSshwOGS6cpjhauSyk5dYAMv4=">AAACCnicdVDLSgMxFM34rPU16tJNaBFclZlpte1CKIjgsoK1hb7IpJk2NJkZkkyhDLN346+4caHi1i9w59+Y6QNU9MCFk3PuJfceN2RUKsv6NFZW19Y3NjNb2e2d3b198+DwTgaRwKSBAxaIloskYdQnDUUVI61QEMRdRpru+DL1mxMiJA38WzUNSZejoU89ipHSUt/MXcEL2PEEwrYDOZz0nOU7DntOEjs86Zt5q1AqVh37HGpSKtrFckrOrGrFgnbBmiEPFqj3zY/OIMARJ77CDEnZtq1QdWMkFMWMJNlOJEmI8BgNSVtTH3Eiu/HslgSeaGUAvUDo8hWcqd8nYsSlnHJXd3KkRvK3l4p/ee1IeZVuTP0wUsTH84+8iEEVwDQYOKCCYMWmmiAsqN4V4hHSQSgdX1aHsLwU/k8aTqFasG5K+Vp1kUYGHIMcOAU2KIMauAZ10AAY3INH8AxejAfjyXg13uatK8Zi5gj8gPH+BUbqmOY=</latexit><latexit sha1_base64="0zBSshwOGS6cpjhauSyk5dYAMv4=">AAACCnicdVDLSgMxFM34rPU16tJNaBFclZlpte1CKIjgsoK1hb7IpJk2NJkZkkyhDLN346+4caHi1i9w59+Y6QNU9MCFk3PuJfceN2RUKsv6NFZW19Y3NjNb2e2d3b198+DwTgaRwKSBAxaIloskYdQnDUUVI61QEMRdRpru+DL1mxMiJA38WzUNSZejoU89ipHSUt/MXcEL2PEEwrYDOZz0nOU7DntOEjs86Zt5q1AqVh37HGpSKtrFckrOrGrFgnbBmiEPFqj3zY/OIMARJ77CDEnZtq1QdWMkFMWMJNlOJEmI8BgNSVtTH3Eiu/HslgSeaGUAvUDo8hWcqd8nYsSlnHJXd3KkRvK3l4p/ee1IeZVuTP0wUsTH84+8iEEVwDQYOKCCYMWmmiAsqN4V4hHSQSgdX1aHsLwU/k8aTqFasG5K+Vp1kUYGHIMcOAU2KIMauAZ10AAY3INH8AxejAfjyXg13uatK8Zi5gj8gPH+BUbqmOY=</latexit>

densità di livelli nello spazio delle fasi

Per passare dal discreto al continuo introduciamo una “densità” nello spazio delle delle fasi:

X

i

!Z

d3~r

Zd3~p g(~r, ~p)

<latexit sha1_base64="hywAQ/vo0in9k/cBOQz8aU61CDo=">AAACJnicdVBdSwJBFJ21L7OvrR57GZLAQGTXtdSXEHrp0SBTcG2ZnR11cPaDmVlBFv9NL/2VXoI+iN76KY26QUYdGDhzzr3ce48bMSqkYXxomZXVtfWN7GZua3tnd0/fP7gVYcwxaeGQhbzjIkEYDUhLUslIJ+IE+S4jbXd0OfPbY8IFDYMbOYlIz0eDgPYpRlJJjn5hi9h3KLRlCG0aSOjdWfaY4IRPl77R1C7CQSG1iql26uh5o1Sx6mXzHCpSsUyrOiNnRr1mQLNkzJEHKZqO/mx7IY59EkjMkBBd04hkL0FcUszINGfHgkQIj9CAdBUNkE9EL5nfOYUnSvFgP+Tqqd3m6s+OBPlCTHxXVfpIDsVvbyb+5XVj2a/1EhpEsSQBXgzqxwyqUGahQY9ygiWbKIIwp2pXiIeIIyxVtDkVwvel8H/SKpfqJeO6km/U0zSy4AgcgwIwQRU0wBVoghbA4B48ghfwqj1oT9qb9r4ozWhpzyFYgvb5BemZpjI=</latexit><latexit sha1_base64="hywAQ/vo0in9k/cBOQz8aU61CDo=">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</latexit><latexit sha1_base64="hywAQ/vo0in9k/cBOQz8aU61CDo=">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</latexit>

Per ovviare a questo problema, introduciamo un volume elementare (arbitrario) nello spazio delle fasi, che, quindi, ha una densità .

Δ1/Δ

Gas perfetto, Maxwell—Boltzmann e temperatura

che, in funzione dell’energia (cinetica), diventa:

g(E)dE =4⇡V m

32

p2E dE

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È importante osservare che la trattazione corretta è quella quantistica. In questo caso si trova che non è arbitrario, ma segue dalla quantizzazione:Δ

costante di Planck� = h3

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Integrando sul volume V occupato dal gas, il numero di punti (livelli) contenuti contenuti nella crosta sferica di raggio p e spessore dp è dato da:

g(~r, ~p)d3~r d3~p ! V

�4⇡p2 dp

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L’energia totale è data da:

funzione di distribuzione di Boltzmann

=3

2

N

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U =3

2N kBT

<latexit sha1_base64="kR6q36wKsVp6ZMDslPYgRzBf8Wo=">AAAB/nicdVDLSsNAFJ3UV62vqODGzWARXEhJmmqbhVB040oqNLbQlDKZTtqhkwczE6HELvwVNy5U3Pod7vwbJ20FFT1w4XDOvdx7jxczKqRhfGi5hcWl5ZX8amFtfWNzS9/euRFRwjFxcMQi3vaQIIyGxJFUMtKOOUGBx0jLG11kfuuWcEGjsCnHMekGaBBSn2IkldTT96ADz6Drc4StMrxyj+Godw6bPb1olCqWXTZPoSIVy7SqGTkx7JoBzZIxRRHM0ejp724/wklAQokZEqJjGrHspohLihmZFNxEkBjhERqQjqIhCojoptP7J/BQKX3oR1xVKOFU/T6RokCIceCpzgDJofjtZeJfXieRfq2b0jBOJAnxbJGfMCgjmIUB+5QTLNlYEYQ5VbdCPEQqC6kiK6gQvj6F/xOnXLJLxnWlWLfnaeTBPjgAR8AEVVAHl6ABHIDBHXgAT+BZu9cetRftddaa0+Yzu+AHtLdPrPqTfw==</latexit><latexit sha1_base64="kR6q36wKsVp6ZMDslPYgRzBf8Wo=">AAAB/nicdVDLSsNAFJ3UV62vqODGzWARXEhJmmqbhVB040oqNLbQlDKZTtqhkwczE6HELvwVNy5U3Pod7vwbJ20FFT1w4XDOvdx7jxczKqRhfGi5hcWl5ZX8amFtfWNzS9/euRFRwjFxcMQi3vaQIIyGxJFUMtKOOUGBx0jLG11kfuuWcEGjsCnHMekGaBBSn2IkldTT96ADz6Drc4StMrxyj+Godw6bPb1olCqWXTZPoSIVy7SqGTkx7JoBzZIxRRHM0ejp724/wklAQokZEqJjGrHspohLihmZFNxEkBjhERqQjqIhCojoptP7J/BQKX3oR1xVKOFU/T6RokCIceCpzgDJofjtZeJfXieRfq2b0jBOJAnxbJGfMCgjmIUB+5QTLNlYEYQ5VbdCPEQqC6kiK6gQvj6F/xOnXLJLxnWlWLfnaeTBPjgAR8AEVVAHl6ABHIDBHXgAT+BZu9cetRftddaa0+Yzu+AHtLdPrPqTfw==</latexit><latexit sha1_base64="kR6q36wKsVp6ZMDslPYgRzBf8Wo=">AAAB/nicdVDLSsNAFJ3UV62vqODGzWARXEhJmmqbhVB040oqNLbQlDKZTtqhkwczE6HELvwVNy5U3Pod7vwbJ20FFT1w4XDOvdx7jxczKqRhfGi5hcWl5ZX8amFtfWNzS9/euRFRwjFxcMQi3vaQIIyGxJFUMtKOOUGBx0jLG11kfuuWcEGjsCnHMekGaBBSn2IkldTT96ADz6Drc4StMrxyj+Godw6bPb1olCqWXTZPoSIVy7SqGTkx7JoBzZIxRRHM0ejp724/wklAQokZEqJjGrHspohLihmZFNxEkBjhERqQjqIhCojoptP7J/BQKX3oR1xVKOFU/T6RokCIceCpzgDJofjtZeJfXieRfq2b0jBOJAnxbJGfMCgjmIUB+5QTLNlYEYQ5VbdCPEQqC6kiK6gQvj6F/xOnXLJLxnWlWLfnaeTBPjgAR8AEVVAHl6ABHIDBHXgAT+BZu9cetRftddaa0+Yzu+AHtLdPrPqTfw==</latexit>

Teoria cinetica:

� =1

kBT<latexit sha1_base64="1JWmcCuzI5jdnBNFyBgpcmi2NMg=">AAACAHicdVDLSsNAFJ3UV62vqBvBzWARXIWkrbZdKEU3Lis0WmhCmEwn7dDJg5mJUELd+CtuXKi49TPc+TdO2goqeuDC4Zx7ufceP2FUSNP80AoLi0vLK8XV0tr6xuaWvr1zLeKUY2LjmMW86yNBGI2ILalkpJtwgkKfkRt/dJH7N7eECxpHHTlOiBuiQUQDipFUkqfvQccnEsFT6AQc4cyaZCPvHHYmnl42jVq1WbFOoCK1qlWt5+TYbDZMaBnmFGUwR9vT351+jNOQRBIzJETPMhPpZohLihmZlJxUkAThERqQnqIRColws+kHE3iolD4MYq4qknCqfp/IUCjEOPRVZ4jkUPz2cvEvr5fKoOFmNEpSSSI8WxSkDMoY5nHAPuUESzZWBGFO1a0QD5FKQqrQSiqEr0/h/8SuGE3DvKqVW2fzNIpgHxyAI2CBOmiBS9AGNsDgDjyAJ/Cs3WuP2ov2OmstaPOZXfAD2tsniymV4Q==</latexit><latexit sha1_base64="1JWmcCuzI5jdnBNFyBgpcmi2NMg=">AAACAHicdVDLSsNAFJ3UV62vqBvBzWARXIWkrbZdKEU3Lis0WmhCmEwn7dDJg5mJUELd+CtuXKi49TPc+TdO2goqeuDC4Zx7ufceP2FUSNP80AoLi0vLK8XV0tr6xuaWvr1zLeKUY2LjmMW86yNBGI2ILalkpJtwgkKfkRt/dJH7N7eECxpHHTlOiBuiQUQDipFUkqfvQccnEsFT6AQc4cyaZCPvHHYmnl42jVq1WbFOoCK1qlWt5+TYbDZMaBnmFGUwR9vT351+jNOQRBIzJETPMhPpZohLihmZlJxUkAThERqQnqIRColws+kHE3iolD4MYq4qknCqfp/IUCjEOPRVZ4jkUPz2cvEvr5fKoOFmNEpSSSI8WxSkDMoY5nHAPuUESzZWBGFO1a0QD5FKQqrQSiqEr0/h/8SuGE3DvKqVW2fzNIpgHxyAI2CBOmiBS9AGNsDgDjyAJ/Cs3WuP2ov2OmstaPOZXfAD2tsniymV4Q==</latexit><latexit sha1_base64="1JWmcCuzI5jdnBNFyBgpcmi2NMg=">AAACAHicdVDLSsNAFJ3UV62vqBvBzWARXIWkrbZdKEU3Lis0WmhCmEwn7dDJg5mJUELd+CtuXKi49TPc+TdO2goqeuDC4Zx7ufceP2FUSNP80AoLi0vLK8XV0tr6xuaWvr1zLeKUY2LjmMW86yNBGI2ILalkpJtwgkKfkRt/dJH7N7eECxpHHTlOiBuiQUQDipFUkqfvQccnEsFT6AQc4cyaZCPvHHYmnl42jVq1WbFOoCK1qlWt5+TYbDZMaBnmFGUwR9vT351+jNOQRBIzJETPMhPpZohLihmZlJxUkAThERqQnqIRColws+kHE3iolD4MYq4qknCqfp/IUCjEOPRVZ4jkUPz2cvEvr5fKoOFmNEpSSSI8WxSkDMoY5nHAPuUESzZWBGFO1a0QD5FKQqrQSiqEr0/h/8SuGE3DvKqVW2fzNIpgHxyAI2CBOmiBS9AGNsDgDjyAJ/Cs3WuP2ov2OmstaPOZXfAD2tsniymV4Q==</latexit>

e, quindi, l’energia totale non dipende da :Δ

U = N

Z 1

0dE

2�p⇡

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Gas perfetto, Maxwell—Boltzmann e temperatura

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dove la funzione di partizione dipende dal volume elementare :Δ

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L’energia totale è data da:

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funzione di distribuzione di Boltzmann

Gas perfetto, Maxwell—Boltzmann e temperatura

0 2 4 6 8 10 12 14

0.1

0.2

0.3

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E (a.u.)<latexit sha1_base64="Imqiyv/w9PB+ss8Pdj1ls/jkjVc=">AAAB83icdVBNS8NAEN3Ur1q/qh69LBahgoSkrba5FUTwWMHYQhPKZrttl+4mcXdTKKG/w4sHFa/+GW/+GzdtBRV9MPB4b4aZeUHMqFSW9WHkVlbX1jfym4Wt7Z3dveL+wZ2MEoGJiyMWiU6AJGE0JK6iipFOLAjiASPtYHyZ+e0JEZJG4a2axsTnaBjSAcVIacm/8s7KnuAQmYl52iuWLLNWdSr2BdSkVrWr9YycW07DgrZpzVECS7R6xXevH+GEk1BhhqTs2las/BQJRTEjs4KXSBIjPEZD0tU0RJxIP50fPYMnWunDQSR0hQrO1e8TKeJSTnmgOzlSI/nby8S/vG6iBg0/pWGcKBLixaJBwqCKYJYA7FNBsGJTTRAWVN8K8QgJhJXOqaBD+PoU/k/ciumY1k2t1HSWaeTBETgGZWCDOmiCa9ACLsDgHjyAJ/BsTIxH48V4XbTmjOXMIfgB4+0TYzaQug==</latexit><latexit sha1_base64="Imqiyv/w9PB+ss8Pdj1ls/jkjVc=">AAAB83icdVBNS8NAEN3Ur1q/qh69LBahgoSkrba5FUTwWMHYQhPKZrttl+4mcXdTKKG/w4sHFa/+GW/+GzdtBRV9MPB4b4aZeUHMqFSW9WHkVlbX1jfym4Wt7Z3dveL+wZ2MEoGJiyMWiU6AJGE0JK6iipFOLAjiASPtYHyZ+e0JEZJG4a2axsTnaBjSAcVIacm/8s7KnuAQmYl52iuWLLNWdSr2BdSkVrWr9YycW07DgrZpzVECS7R6xXevH+GEk1BhhqTs2las/BQJRTEjs4KXSBIjPEZD0tU0RJxIP50fPYMnWunDQSR0hQrO1e8TKeJSTnmgOzlSI/nby8S/vG6iBg0/pWGcKBLixaJBwqCKYJYA7FNBsGJTTRAWVN8K8QgJhJXOqaBD+PoU/k/ciumY1k2t1HSWaeTBETgGZWCDOmiCa9ACLsDgHjyAJ/BsTIxH48V4XbTmjOXMIfgB4+0TYzaQug==</latexit><latexit sha1_base64="Imqiyv/w9PB+ss8Pdj1ls/jkjVc=">AAAB83icdVBNS8NAEN3Ur1q/qh69LBahgoSkrba5FUTwWMHYQhPKZrttl+4mcXdTKKG/w4sHFa/+GW/+GzdtBRV9MPB4b4aZeUHMqFSW9WHkVlbX1jfym4Wt7Z3dveL+wZ2MEoGJiyMWiU6AJGE0JK6iipFOLAjiASPtYHyZ+e0JEZJG4a2axsTnaBjSAcVIacm/8s7KnuAQmYl52iuWLLNWdSr2BdSkVrWr9YycW07DgrZpzVECS7R6xXevH+GEk1BhhqTs2las/BQJRTEjs4KXSBIjPEZD0tU0RJxIP50fPYMnWunDQSR0hQrO1e8TKeJSTnmgOzlSI/nby8S/vG6iBg0/pWGcKBLixaJBwqCKYJYA7FNBsGJTTRAWVN8K8QgJhJXOqaBD+PoU/k/ciumY1k2t1HSWaeTBETgGZWCDOmiCa9ACLsDgHjyAJ/BsTIxH48V4XbTmjOXMIfgB4+0TYzaQug==</latexit>

� = 0.1<latexit sha1_base64="PtYC0aOl0tu2PyvdjpRuHdsTDVU=">AAAB8XicdVDLSgMxFM34rPVVdekmWARXQ6YdbbsQCm5cVnBsYTqUTJq2oZkHyR2hlH6GGxcqbv0bd/6NmbaCih4InJxzL/feE6ZSaCDkw1pZXVvf2CxsFbd3dvf2SweHdzrJFOMeS2SiOiHVXIqYeyBA8k6qOI1Cydvh+Cr32/dcaZHEtzBJeRDRYSwGglEwkt8NOVB8iYnt9EplYrvVRsW5MF/XrTrVWk7OSaNOsGOTOcpoiVav9N7tJyyLeAxMUq19h6QQTKkCwSSfFbuZ5illYzrkvqExjbgOpvOVZ/jUKH08SJR5MeC5+r1jSiOtJ1FoKiMKI/3by8W/PD+DQT2YijjNgMdsMWiQSQwJzu/HfaE4AzkxhDIlzK6YjaiiDExKRRPC16X4f+JV7IZNbtxy012mUUDH6ASdIQfVUBNdoxbyEEMJekBP6NkC69F6sV4XpSvWsucI/YD19gkWxY/6</latexit><latexit sha1_base64="PtYC0aOl0tu2PyvdjpRuHdsTDVU=">AAAB8XicdVDLSgMxFM34rPVVdekmWARXQ6YdbbsQCm5cVnBsYTqUTJq2oZkHyR2hlH6GGxcqbv0bd/6NmbaCih4InJxzL/feE6ZSaCDkw1pZXVvf2CxsFbd3dvf2SweHdzrJFOMeS2SiOiHVXIqYeyBA8k6qOI1Cydvh+Cr32/dcaZHEtzBJeRDRYSwGglEwkt8NOVB8iYnt9EplYrvVRsW5MF/XrTrVWk7OSaNOsGOTOcpoiVav9N7tJyyLeAxMUq19h6QQTKkCwSSfFbuZ5illYzrkvqExjbgOpvOVZ/jUKH08SJR5MeC5+r1jSiOtJ1FoKiMKI/3by8W/PD+DQT2YijjNgMdsMWiQSQwJzu/HfaE4AzkxhDIlzK6YjaiiDExKRRPC16X4f+JV7IZNbtxy012mUUDH6ASdIQfVUBNdoxbyEEMJekBP6NkC69F6sV4XpSvWsucI/YD19gkWxY/6</latexit><latexit sha1_base64="PtYC0aOl0tu2PyvdjpRuHdsTDVU=">AAAB8XicdVDLSgMxFM34rPVVdekmWARXQ6YdbbsQCm5cVnBsYTqUTJq2oZkHyR2hlH6GGxcqbv0bd/6NmbaCih4InJxzL/feE6ZSaCDkw1pZXVvf2CxsFbd3dvf2SweHdzrJFOMeS2SiOiHVXIqYeyBA8k6qOI1Cydvh+Cr32/dcaZHEtzBJeRDRYSwGglEwkt8NOVB8iYnt9EplYrvVRsW5MF/XrTrVWk7OSaNOsGOTOcpoiVav9N7tJyyLeAxMUq19h6QQTKkCwSSfFbuZ5illYzrkvqExjbgOpvOVZ/jUKH08SJR5MeC5+r1jSiOtJ1FoKiMKI/3by8W/PD+DQT2YijjNgMdsMWiQSQwJzu/HfaE4AzkxhDIlzK6YjaiiDExKRRPC16X4f+JV7IZNbtxy012mUUDH6ASdIQfVUBNdoxbyEEMJekBP6NkC69F6sV4XpSvWsucI/YD19gkWxY/6</latexit>

� = 0.2<latexit sha1_base64="x3MZuGyZ3vHnjGjH7oVQrjqWiAA=">AAAB8XicdVDLSgMxFM34rPVVdekmWARXQ6YdbbsQCm5cVrC2MB1KJs20oZkHyR2hDP0MNy5U3Po37vwb04egogcCJ+fcy733BKkUGgj5sFZW19Y3Ngtbxe2d3b390sHhnU4yxXibJTJR3YBqLkXM2yBA8m6qOI0CyTvB+Grmd+650iKJb2GScj+iw1iEglEwktcLOFB8iYld6ZfKxHarjYpzYb6uW3WqtRk5J406wY5N5iijJVr90ntvkLAs4jEwSbX2HJKCn1MFgkk+LfYyzVPKxnTIPUNjGnHt5/OVp/jUKAMcJsq8GPBc/d6R00jrSRSYyojCSP/2ZuJfnpdBWPdzEacZ8JgtBoWZxJDg2f14IBRnICeGUKaE2RWzEVWUgUmpaEL4uhT/T9oVu2GTG7fcdJdpFNAxOkFnyEE11ETXqIXaiKEEPaAn9GyB9Wi9WK+L0hVr2XOEfsB6+wQYSI/7</latexit><latexit sha1_base64="x3MZuGyZ3vHnjGjH7oVQrjqWiAA=">AAAB8XicdVDLSgMxFM34rPVVdekmWARXQ6YdbbsQCm5cVrC2MB1KJs20oZkHyR2hDP0MNy5U3Po37vwb04egogcCJ+fcy733BKkUGgj5sFZW19Y3Ngtbxe2d3b390sHhnU4yxXibJTJR3YBqLkXM2yBA8m6qOI0CyTvB+Grmd+650iKJb2GScj+iw1iEglEwktcLOFB8iYld6ZfKxHarjYpzYb6uW3WqtRk5J406wY5N5iijJVr90ntvkLAs4jEwSbX2HJKCn1MFgkk+LfYyzVPKxnTIPUNjGnHt5/OVp/jUKAMcJsq8GPBc/d6R00jrSRSYyojCSP/2ZuJfnpdBWPdzEacZ8JgtBoWZxJDg2f14IBRnICeGUKaE2RWzEVWUgUmpaEL4uhT/T9oVu2GTG7fcdJdpFNAxOkFnyEE11ETXqIXaiKEEPaAn9GyB9Wi9WK+L0hVr2XOEfsB6+wQYSI/7</latexit><latexit sha1_base64="x3MZuGyZ3vHnjGjH7oVQrjqWiAA=">AAAB8XicdVDLSgMxFM34rPVVdekmWARXQ6YdbbsQCm5cVrC2MB1KJs20oZkHyR2hDP0MNy5U3Po37vwb04egogcCJ+fcy733BKkUGgj5sFZW19Y3Ngtbxe2d3b390sHhnU4yxXibJTJR3YBqLkXM2yBA8m6qOI0CyTvB+Grmd+650iKJb2GScj+iw1iEglEwktcLOFB8iYld6ZfKxHarjYpzYb6uW3WqtRk5J406wY5N5iijJVr90ntvkLAs4jEwSbX2HJKCn1MFgkk+LfYyzVPKxnTIPUNjGnHt5/OVp/jUKAMcJsq8GPBc/d6R00jrSRSYyojCSP/2ZuJfnpdBWPdzEacZ8JgtBoWZxJDg2f14IBRnICeGUKaE2RWzEVWUgUmpaEL4uhT/T9oVu2GTG7fcdJdpFNAxOkFnyEE11ETXqIXaiKEEPaAn9GyB9Wi9WK+L0hVr2XOEfsB6+wQYSI/7</latexit>

� = 1.0<latexit sha1_base64="fWrE+72QwKTU2hOLlj7tOHIgT2I=">AAAB8XicdVDLSsNAFJ34rPVVdelmsAiuQtJG2y6EghuXFYwtpKFMppN26OTBzI1QQj/DjQsVt/6NO//GSVtBRQ8MHM45l7n3BKngCizrw1hZXVvf2Cxtlbd3dvf2KweHdyrJJGUuTUQiewFRTPCYucBBsF4qGYkCwbrB5Krwu/dMKp7EtzBNmR+RUcxDTgloyesHDAi+xLZpDSpVy3TqrZp9gTVx6na9UZBzq9W0isAcVbREZ1B57w8TmkUsBiqIUp5tpeDnRAKngs3K/UyxlNAJGTFP05hETPn5fOUZPtXKEIeJ1C8GPFe/T+QkUmoaBToZERir314h/uV5GYRNP+dxmgGL6eKjMBMYElzcj4dcMgpiqgmhkutdMR0TSSjolsq6hK9L8f/ErZkt07pxqm1n2UYJHaMTdIZs1EBtdI06yEUUJegBPaFnA4xH48V4XURXjOXMEfoB4+0TFseP+g==</latexit><latexit sha1_base64="fWrE+72QwKTU2hOLlj7tOHIgT2I=">AAAB8XicdVDLSsNAFJ34rPVVdelmsAiuQtJG2y6EghuXFYwtpKFMppN26OTBzI1QQj/DjQsVt/6NO//GSVtBRQ8MHM45l7n3BKngCizrw1hZXVvf2Cxtlbd3dvf2KweHdyrJJGUuTUQiewFRTPCYucBBsF4qGYkCwbrB5Krwu/dMKp7EtzBNmR+RUcxDTgloyesHDAi+xLZpDSpVy3TqrZp9gTVx6na9UZBzq9W0isAcVbREZ1B57w8TmkUsBiqIUp5tpeDnRAKngs3K/UyxlNAJGTFP05hETPn5fOUZPtXKEIeJ1C8GPFe/T+QkUmoaBToZERir314h/uV5GYRNP+dxmgGL6eKjMBMYElzcj4dcMgpiqgmhkutdMR0TSSjolsq6hK9L8f/ErZkt07pxqm1n2UYJHaMTdIZs1EBtdI06yEUUJegBPaFnA4xH48V4XURXjOXMEfoB4+0TFseP+g==</latexit><latexit sha1_base64="fWrE+72QwKTU2hOLlj7tOHIgT2I=">AAAB8XicdVDLSsNAFJ34rPVVdelmsAiuQtJG2y6EghuXFYwtpKFMppN26OTBzI1QQj/DjQsVt/6NO//GSVtBRQ8MHM45l7n3BKngCizrw1hZXVvf2Cxtlbd3dvf2KweHdyrJJGUuTUQiewFRTPCYucBBsF4qGYkCwbrB5Krwu/dMKp7EtzBNmR+RUcxDTgloyesHDAi+xLZpDSpVy3TqrZp9gTVx6na9UZBzq9W0isAcVbREZ1B57w8TmkUsBiqIUp5tpeDnRAKngs3K/UyxlNAJGTFP05hETPn5fOUZPtXKEIeJ1C8GPFe/T+QkUmoaBToZERir314h/uV5GYRNP+dxmgGL6eKjMBMYElzcj4dcMgpiqgmhkutdMR0TSSjolsq6hK9L8f/ErZkt07pxqm1n2UYJHaMTdIZs1EBtdI06yEUUJegBPaFnA4xH48V4XURXjOXMEfoB4+0TFseP+g==</latexit>