Bose-Einstein distribution N =5balitsky.com/teaching/phys454/hw454_11sol.pdfFermi-Dirac distribution...

3
Bose-Einstein distribution 4 5 2 1 0 3 45 90 90 108 108 90 21 0.08 0.16 0.36 0.72 1.43 2.25 = 552 ¯ N j N =5, E =5W = Q j (g j +N j -1)! (g j -1)!N j ! = Q j (N j +2)! 2N j ! W

Transcript of Bose-Einstein distribution N =5balitsky.com/teaching/phys454/hw454_11sol.pdfFermi-Dirac distribution...

Bose-Einstein distribution

4

5

2

1

0

3

45 90 90 108 108 90 21

0.08

0.16

0.36

0.72

1.43

2.25

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N = 5, E = 5✏<latexit sha1_base64="sj+76Dki89YSyL26Lz+D4UEf1us=">AAAB/XicdZDLSgMxFIbPeK31VhVXboJFcCFlRqzahVAUwZVUcGyhHUomTdvQzIUkI5Shg6/ixoWKW9/DnW9jph2hih4I+fn+c8jJ74acSWWan8bM7Nz8wmJuKb+8srq2XtjYvJNBJAi1ScAD0XCxpJz51FZMcdoIBcWey2ndHVykfv2eCskC/1YNQ+p4uOezLiNYadQubF8nZ0n5IEku07tFQ8l4yotmyRwXmhJl06ocW8jKSBGyqrULH61OQCKP+opwLGXTMkPlxFgoRjgd5VuRpCEmA9yjTS197FHpxOP1R2hPkw7qBkIfX6ExnZ6IsSfl0HN1p4dVX/72UviX14xU99SJmR9Givpk8lA34kgFKM0CdZigRPGhFpgIpndFpI8FJkonltchfP8U/S/sw1KlZN4cFavnWRo52IFd2AcLTqAKV1ADGwjE8AjP8GI8GE/Gq/E2aZ0xspkt+FHG+xdhU5VT</latexit><latexit sha1_base64="sj+76Dki89YSyL26Lz+D4UEf1us=">AAAB/XicdZDLSgMxFIbPeK31VhVXboJFcCFlRqzahVAUwZVUcGyhHUomTdvQzIUkI5Shg6/ixoWKW9/DnW9jph2hih4I+fn+c8jJ74acSWWan8bM7Nz8wmJuKb+8srq2XtjYvJNBJAi1ScAD0XCxpJz51FZMcdoIBcWey2ndHVykfv2eCskC/1YNQ+p4uOezLiNYadQubF8nZ0n5IEku07tFQ8l4yotmyRwXmhJl06ocW8jKSBGyqrULH61OQCKP+opwLGXTMkPlxFgoRjgd5VuRpCEmA9yjTS197FHpxOP1R2hPkw7qBkIfX6ExnZ6IsSfl0HN1p4dVX/72UviX14xU99SJmR9Givpk8lA34kgFKM0CdZigRPGhFpgIpndFpI8FJkonltchfP8U/S/sw1KlZN4cFavnWRo52IFd2AcLTqAKV1ADGwjE8AjP8GI8GE/Gq/E2aZ0xspkt+FHG+xdhU5VT</latexit><latexit sha1_base64="sj+76Dki89YSyL26Lz+D4UEf1us=">AAAB/XicdZDLSgMxFIbPeK31VhVXboJFcCFlRqzahVAUwZVUcGyhHUomTdvQzIUkI5Shg6/ixoWKW9/DnW9jph2hih4I+fn+c8jJ74acSWWan8bM7Nz8wmJuKb+8srq2XtjYvJNBJAi1ScAD0XCxpJz51FZMcdoIBcWey2ndHVykfv2eCskC/1YNQ+p4uOezLiNYadQubF8nZ0n5IEku07tFQ8l4yotmyRwXmhJl06ocW8jKSBGyqrULH61OQCKP+opwLGXTMkPlxFgoRjgd5VuRpCEmA9yjTS197FHpxOP1R2hPkw7qBkIfX6ExnZ6IsSfl0HN1p4dVX/72UviX14xU99SJmR9Givpk8lA34kgFKM0CdZigRPGhFpgIpndFpI8FJkonltchfP8U/S/sw1KlZN4cFavnWRo52IFd2AcLTqAKV1ADGwjE8AjP8GI8GE/Gq/E2aZ0xspkt+FHG+xdhU5VT</latexit>

W =Q

j(gj+Nj�1)!(gj�1)!Nj !

=Q

j(Nj+2)!2Nj !

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Maxwell-Boltzmann distribution

N̄j<latexit sha1_base64="x17RpWkQGVieIE6n2LXddq5e764=">AAAB73icdVBNSwMxEJ31s9avqkcvwSJ4KrviV29FL56kgmsr7VKyabaNTbJLkhXK0l/hxYOKV/+ON/+NabtCFX0w8Hhvhpl5YcKZNq776czNLywuLRdWiqtr6xubpa3tWx2nilCfxDxWzRBrypmkvmGG02aiKBYhp41wcDH2Gw9UaRbLGzNMaCBwT7KIEWysdNcOscquRp37TqnsVtwJ0Aw5dr3qiYe8XClDjnqn9NHuxiQVVBrCsdYtz01MkGFlGOF0VGynmiaYDHCPtiyVWFAdZJODR2jfKl0UxcqWNGiizk5kWGg9FKHtFNj09W9vLP7ltVITnQUZk0lqqCTTRVHKkYnR+HvUZYoSw4eWYKKYvRWRPlaYGJtR0Ybw/Sn6n/iHlWrFvT4q187zNAqwC3twAB6cQg0uoQ4+EBDwCM/w4ijnyXl13qatc04+swM/4Lx/AW6kkF0=</latexit><latexit sha1_base64="x17RpWkQGVieIE6n2LXddq5e764=">AAAB73icdVBNSwMxEJ31s9avqkcvwSJ4KrviV29FL56kgmsr7VKyabaNTbJLkhXK0l/hxYOKV/+ON/+NabtCFX0w8Hhvhpl5YcKZNq776czNLywuLRdWiqtr6xubpa3tWx2nilCfxDxWzRBrypmkvmGG02aiKBYhp41wcDH2Gw9UaRbLGzNMaCBwT7KIEWysdNcOscquRp37TqnsVtwJ0Aw5dr3qiYe8XClDjnqn9NHuxiQVVBrCsdYtz01MkGFlGOF0VGynmiaYDHCPtiyVWFAdZJODR2jfKl0UxcqWNGiizk5kWGg9FKHtFNj09W9vLP7ltVITnQUZk0lqqCTTRVHKkYnR+HvUZYoSw4eWYKKYvRWRPlaYGJtR0Ybw/Sn6n/iHlWrFvT4q187zNAqwC3twAB6cQg0uoQ4+EBDwCM/w4ijnyXl13qatc04+swM/4Lx/AW6kkF0=</latexit><latexit sha1_base64="x17RpWkQGVieIE6n2LXddq5e764=">AAAB73icdVBNSwMxEJ31s9avqkcvwSJ4KrviV29FL56kgmsr7VKyabaNTbJLkhXK0l/hxYOKV/+ON/+NabtCFX0w8Hhvhpl5YcKZNq776czNLywuLRdWiqtr6xubpa3tWx2nilCfxDxWzRBrypmkvmGG02aiKBYhp41wcDH2Gw9UaRbLGzNMaCBwT7KIEWysdNcOscquRp37TqnsVtwJ0Aw5dr3qiYe8XClDjnqn9NHuxiQVVBrCsdYtz01MkGFlGOF0VGynmiaYDHCPtiyVWFAdZJODR2jfKl0UxcqWNGiizk5kWGg9FKHtFNj09W9vLP7ltVITnQUZk0lqqCTTRVHKkYnR+HvUZYoSw4eWYKKYvRWRPlaYGJtR0Ybw/Sn6n/iHlWrFvT4q187zNAqwC3twAB6cQg0uoQ4+EBDwCM/w4ijnyXl13qatc04+swM/4Lx/AW6kkF0=</latexit>

W ⇥ 3�5<latexit sha1_base64="I1/iGtYZphk7+1YSREMACYgsASU=">AAAB/XicdVBNS8NAEJ3Ur1q/ouLJy2IRvFgStWpvRS8eKxhbaGLZbLft0s0HuxuhhIB/xYsHFa/+D2/+G7dthCr6YODx3gwz8/yYM6ks69MozM0vLC4Vl0srq2vrG+bm1q2MEkGoQyIeiZaPJeUspI5iitNWLCgOfE6b/vBy7DfvqZAsCm/UKKZegPsh6zGClZY65k7qEsxRM3MVC6hEx3fpYTXrmGWrYk2AZkjVsmunNrJzpQw5Gh3zw+1GJAloqAjHUrZtK1ZeioVihNOs5CaSxpgMcZ+2NQ2x3uWlk/MztK+VLupFQleo0ESdnUhxIOUo8HVngNVA/vbG4l9eO1G9cy9lYZwoGpLpol7CkYrQOAvUZYISxUeaYCKYvhWRARaYKJ1YSYfw/Sn6nzhHlVrFuj4p1y/yNIqwC3twADacQR2uoAEOEEjhEZ7hxXgwnoxX423aWjDymW34AeP9C8UUlOo=</latexit><latexit sha1_base64="I1/iGtYZphk7+1YSREMACYgsASU=">AAAB/XicdVBNS8NAEJ3Ur1q/ouLJy2IRvFgStWpvRS8eKxhbaGLZbLft0s0HuxuhhIB/xYsHFa/+D2/+G7dthCr6YODx3gwz8/yYM6ks69MozM0vLC4Vl0srq2vrG+bm1q2MEkGoQyIeiZaPJeUspI5iitNWLCgOfE6b/vBy7DfvqZAsCm/UKKZegPsh6zGClZY65k7qEsxRM3MVC6hEx3fpYTXrmGWrYk2AZkjVsmunNrJzpQw5Gh3zw+1GJAloqAjHUrZtK1ZeioVihNOs5CaSxpgMcZ+2NQ2x3uWlk/MztK+VLupFQleo0ESdnUhxIOUo8HVngNVA/vbG4l9eO1G9cy9lYZwoGpLpol7CkYrQOAvUZYISxUeaYCKYvhWRARaYKJ1YSYfw/Sn6nzhHlVrFuj4p1y/yNIqwC3twADacQR2uoAEOEEjhEZ7hxXgwnoxX423aWjDymW34AeP9C8UUlOo=</latexit><latexit sha1_base64="I1/iGtYZphk7+1YSREMACYgsASU=">AAAB/XicdVBNS8NAEJ3Ur1q/ouLJy2IRvFgStWpvRS8eKxhbaGLZbLft0s0HuxuhhIB/xYsHFa/+D2/+G7dthCr6YODx3gwz8/yYM6ks69MozM0vLC4Vl0srq2vrG+bm1q2MEkGoQyIeiZaPJeUspI5iitNWLCgOfE6b/vBy7DfvqZAsCm/UKKZegPsh6zGClZY65k7qEsxRM3MVC6hEx3fpYTXrmGWrYk2AZkjVsmunNrJzpQw5Gh3zw+1GJAloqAjHUrZtK1ZeioVihNOs5CaSxpgMcZ+2NQ2x3uWlk/MztK+VLupFQleo0ESdnUhxIOUo8HVngNVA/vbG4l9eO1G9cy9lYZwoGpLpol7CkYrQOAvUZYISxUeaYCKYvhWRARaYKJ1YSYfw/Sn6nzhHlVrFuj4p1y/yNIqwC3twADacQR2uoAEOEEjhEZ7hxXgwnoxX423aWjDymW34AeP9C8UUlOo=</latexit>

W = N !Y

j

gNj

j

Nj !

= 5!Y

j

3Nj

Nj !

=355!QNj !

<latexit sha1_base64="obMzM+m6F7RkJgxOECzUr/qY3Jw=">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</latexit><latexit sha1_base64="obMzM+m6F7RkJgxOECzUr/qY3Jw=">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</latexit><latexit sha1_base64="obMzM+m6F7RkJgxOECzUr/qY3Jw=">AAACbXicbVHbSsMwGE7reZ42BS9UJDpEr0arzsPFQPTGq6HgnLDOkmbZli1tSpIKo7R3vqB3PoM3PoJZNzzuh8CX7/An+eOFjEplWW+GOTU9Mzs3v5BbXFpeWc0X1h4kjwQmNcwZF48ekoTRgNQUVYw8hoIg32Ok7vWvh3r9mQhJeXCvBiFp+qgT0DbFSGnKzb/ETtYk9lhEEr1BDNaTtJJWd51Q8Jbbiztu7ymuur3E4boT1Gg3SRwn9ytZSctfgeOJ9kqqhbJ2DenMOtJSXW6+aJXKln1xakOrZGX1DewxKIJx3br5V6fFceSTQGGGpGzYVqiaMRKKYkaSnBNJEiLcRx3S0DBAPpHNOLtvAvc104JtLvQKFMzYn4kY+VIOfE87faS68q82JCdpjUi1z5sxDcJIkQCPDmpHDCoOh7OHLSoIVmygAcKC6rtC3EUCYaV/KKeHYP998n9QOypdlKy7k+Ll1Xga82AL7IFDYIMzcAluwC2oAQzejYKxaWwZH+aGuW3ujKymMc6sg19lHnwCG2a9NA==</latexit>

⌦ = 126⇥ 35<latexit sha1_base64="5VAmCNoDxkNWPX42HRW6+9FtBL8=">AAAB/nicdVBNS8NAEN3Ur1q/ooIXL4tF8FSSaqs9CEUv3qxgbKGJZbPdtEs3m7C7EUptwb/ixYOKV3+HN/+N2zZCFX0w8Hhvhpl5fsyoVJb1aWTm5hcWl7LLuZXVtfUNc3PrRkaJwMTBEYtEw0eSMMqJo6hipBELgkKfkbrfOx/79TsiJI34terHxAtRh9OAYqS01DJ33MuQdNDodGQXy66iIZHw8LbUMvNWwZoAzpCSZVfKNrRTJQ9S1Frmh9uOcBISrjBDUjZtK1beAAlFMSPDnJtIEiPcQx3S1JQjvcgbTO4fwn2ttGEQCV1cwYk6OzFAoZT90NedIVJd+dsbi395zUQFJ96A8jhRhOPpoiBhUEVwHAZsU0GwYn1NEBZU3wpxFwmElY4sp0P4/hT+T5xioVKwro7y1bM0jSzYBXvgANjgGFTBBagBB2BwDx7BM3gxHown49V4m7ZmjHRmG/yA8f4F4QCU6w==</latexit><latexit sha1_base64="5VAmCNoDxkNWPX42HRW6+9FtBL8=">AAAB/nicdVBNS8NAEN3Ur1q/ooIXL4tF8FSSaqs9CEUv3qxgbKGJZbPdtEs3m7C7EUptwb/ixYOKV3+HN/+N2zZCFX0w8Hhvhpl5fsyoVJb1aWTm5hcWl7LLuZXVtfUNc3PrRkaJwMTBEYtEw0eSMMqJo6hipBELgkKfkbrfOx/79TsiJI34terHxAtRh9OAYqS01DJ33MuQdNDodGQXy66iIZHw8LbUMvNWwZoAzpCSZVfKNrRTJQ9S1Frmh9uOcBISrjBDUjZtK1beAAlFMSPDnJtIEiPcQx3S1JQjvcgbTO4fwn2ttGEQCV1cwYk6OzFAoZT90NedIVJd+dsbi395zUQFJ96A8jhRhOPpoiBhUEVwHAZsU0GwYn1NEBZU3wpxFwmElY4sp0P4/hT+T5xioVKwro7y1bM0jSzYBXvgANjgGFTBBagBB2BwDx7BM3gxHown49V4m7ZmjHRmG/yA8f4F4QCU6w==</latexit><latexit sha1_base64="5VAmCNoDxkNWPX42HRW6+9FtBL8=">AAAB/nicdVBNS8NAEN3Ur1q/ooIXL4tF8FSSaqs9CEUv3qxgbKGJZbPdtEs3m7C7EUptwb/ixYOKV3+HN/+N2zZCFX0w8Hhvhpl5fsyoVJb1aWTm5hcWl7LLuZXVtfUNc3PrRkaJwMTBEYtEw0eSMMqJo6hipBELgkKfkbrfOx/79TsiJI34terHxAtRh9OAYqS01DJ33MuQdNDodGQXy66iIZHw8LbUMvNWwZoAzpCSZVfKNrRTJQ9S1Frmh9uOcBISrjBDUjZtK1beAAlFMSPDnJtIEiPcQx3S1JQjvcgbTO4fwn2ttGEQCV1cwYk6OzFAoZT90NedIVJd+dsbi395zUQFJ96A8jhRhOPpoiBhUEVwHAZsU0GwYn1NEBZU3wpxFwmElY4sp0P4/hT+T5xioVKwro7y1bM0jSzYBXvgANjgGFTBBagBB2BwDx7BM3gxHown49V4m7ZmjHRmG/yA8f4F4QCU6w==</latexit>

4

5

2

1

0

3

5 20 20 30 30 20 1

0.04

0.16

0.40

0.80

1.39

2.22

N = 5, E = 5✏<latexit sha1_base64="sj+76Dki89YSyL26Lz+D4UEf1us=">AAAB/XicdZDLSgMxFIbPeK31VhVXboJFcCFlRqzahVAUwZVUcGyhHUomTdvQzIUkI5Shg6/ixoWKW9/DnW9jph2hih4I+fn+c8jJ74acSWWan8bM7Nz8wmJuKb+8srq2XtjYvJNBJAi1ScAD0XCxpJz51FZMcdoIBcWey2ndHVykfv2eCskC/1YNQ+p4uOezLiNYadQubF8nZ0n5IEku07tFQ8l4yotmyRwXmhJl06ocW8jKSBGyqrULH61OQCKP+opwLGXTMkPlxFgoRjgd5VuRpCEmA9yjTS197FHpxOP1R2hPkw7qBkIfX6ExnZ6IsSfl0HN1p4dVX/72UviX14xU99SJmR9Givpk8lA34kgFKM0CdZigRPGhFpgIpndFpI8FJkonltchfP8U/S/sw1KlZN4cFavnWRo52IFd2AcLTqAKV1ADGwjE8AjP8GI8GE/Gq/E2aZ0xspkt+FHG+xdhU5VT</latexit><latexit sha1_base64="sj+76Dki89YSyL26Lz+D4UEf1us=">AAAB/XicdZDLSgMxFIbPeK31VhVXboJFcCFlRqzahVAUwZVUcGyhHUomTdvQzIUkI5Shg6/ixoWKW9/DnW9jph2hih4I+fn+c8jJ74acSWWan8bM7Nz8wmJuKb+8srq2XtjYvJNBJAi1ScAD0XCxpJz51FZMcdoIBcWey2ndHVykfv2eCskC/1YNQ+p4uOezLiNYadQubF8nZ0n5IEku07tFQ8l4yotmyRwXmhJl06ocW8jKSBGyqrULH61OQCKP+opwLGXTMkPlxFgoRjgd5VuRpCEmA9yjTS197FHpxOP1R2hPkw7qBkIfX6ExnZ6IsSfl0HN1p4dVX/72UviX14xU99SJmR9Givpk8lA34kgFKM0CdZigRPGhFpgIpndFpI8FJkonltchfP8U/S/sw1KlZN4cFavnWRo52IFd2AcLTqAKV1ADGwjE8AjP8GI8GE/Gq/E2aZ0xspkt+FHG+xdhU5VT</latexit><latexit sha1_base64="sj+76Dki89YSyL26Lz+D4UEf1us=">AAAB/XicdZDLSgMxFIbPeK31VhVXboJFcCFlRqzahVAUwZVUcGyhHUomTdvQzIUkI5Shg6/ixoWKW9/DnW9jph2hih4I+fn+c8jJ74acSWWan8bM7Nz8wmJuKb+8srq2XtjYvJNBJAi1ScAD0XCxpJz51FZMcdoIBcWey2ndHVykfv2eCskC/1YNQ+p4uOezLiNYadQubF8nZ0n5IEku07tFQ8l4yotmyRwXmhJl06ocW8jKSBGyqrULH61OQCKP+opwLGXTMkPlxFgoRjgd5VuRpCEmA9yjTS197FHpxOP1R2hPkw7qBkIfX6ExnZ6IsSfl0HN1p4dVX/72UviX14xU99SJmR9Givpk8lA34kgFKM0CdZigRPGhFpgIpndFpI8FJkonltchfP8U/S/sw1KlZN4cFavnWRo52IFd2AcLTqAKV1ADGwjE8AjP8GI8GE/Gq/E2aZ0xspkt+FHG+xdhU5VT</latexit>

Fermi-Dirac distribution N = 7, E = 7✏<latexit sha1_base64="wCGReSX3AhxJTKuRwX0KIF3X4dU=">AAAB/nicdVBNS8NAEN3Ur1q/qoIXL4tF8CAlEbX2IBRF8CQVjC20oWy2m3bpZhN2N0KJDfhXvHhQ8erv8Oa/cdNGqKIPBh7vzTAzzw0Zlco0P43czOzc/EJ+sbC0vLK6VlzfuJVBJDCxccAC0XSRJIxyYiuqGGmGgiDfZaThDs5Tv3FHhKQBv1HDkDg+6nHqUYyUljrFravkNKnsJ0lykZI2CSVlqVEyy+YYcIocmVb12IJWppRAhnqn+NHuBjjyCVeYISlblhkqJ0ZCUczIqNCOJAkRHqAeaWnKkU+kE4/vH8FdrXShFwhdXMGxOj0RI1/Koe/qTh+pvvztpeJfXitS3okTUx5GinA8WeRFDKoApmHALhUEKzbUBGFB9a0Q95FAWOnICjqE70/h/8Q+KFfL5vVhqXaWpZEH22AH7AELVEANXII6sAEG9+ARPIMX48F4Ml6Nt0lrzshmNsEPGO9fVI6V3w==</latexit><latexit sha1_base64="wCGReSX3AhxJTKuRwX0KIF3X4dU=">AAAB/nicdVBNS8NAEN3Ur1q/qoIXL4tF8CAlEbX2IBRF8CQVjC20oWy2m3bpZhN2N0KJDfhXvHhQ8erv8Oa/cdNGqKIPBh7vzTAzzw0Zlco0P43czOzc/EJ+sbC0vLK6VlzfuJVBJDCxccAC0XSRJIxyYiuqGGmGgiDfZaThDs5Tv3FHhKQBv1HDkDg+6nHqUYyUljrFravkNKnsJ0lykZI2CSVlqVEyy+YYcIocmVb12IJWppRAhnqn+NHuBjjyCVeYISlblhkqJ0ZCUczIqNCOJAkRHqAeaWnKkU+kE4/vH8FdrXShFwhdXMGxOj0RI1/Koe/qTh+pvvztpeJfXitS3okTUx5GinA8WeRFDKoApmHALhUEKzbUBGFB9a0Q95FAWOnICjqE70/h/8Q+KFfL5vVhqXaWpZEH22AH7AELVEANXII6sAEG9+ARPIMX48F4Ml6Nt0lrzshmNsEPGO9fVI6V3w==</latexit><latexit sha1_base64="wCGReSX3AhxJTKuRwX0KIF3X4dU=">AAAB/nicdVBNS8NAEN3Ur1q/qoIXL4tF8CAlEbX2IBRF8CQVjC20oWy2m3bpZhN2N0KJDfhXvHhQ8erv8Oa/cdNGqKIPBh7vzTAzzw0Zlco0P43czOzc/EJ+sbC0vLK6VlzfuJVBJDCxccAC0XSRJIxyYiuqGGmGgiDfZaThDs5Tv3FHhKQBv1HDkDg+6nHqUYyUljrFravkNKnsJ0lykZI2CSVlqVEyy+YYcIocmVb12IJWppRAhnqn+NHuBjjyCVeYISlblhkqJ0ZCUczIqNCOJAkRHqAeaWnKkU+kE4/vH8FdrXShFwhdXMGxOj0RI1/Koe/qTh+pvvztpeJfXitS3okTUx5GinA8WeRFDKoApmHALhUEKzbUBGFB9a0Q95FAWOnICjqE70/h/8Q+KFfL5vVhqXaWpZEH22AH7AELVEANXII6sAEG9+ARPIMX48F4Ml6Nt0lrzshmNsEPGO9fVI6V3w==</latexit>

⌦ = 42<latexit sha1_base64="brUkHsmwbjExLPrfm3Bvs8RQTcI=">AAAB8XicdVDLSgNBEJyNrxhfUY9eBoPgKeyG+MhBCHrxZgTXBDZLmJ3MJkPmsczMCiEkf+HFg4pX/8abf+MkWSGKFjQUVd10d0UJo9q47qeTW1peWV3Lrxc2Nre2d4q7e/dapgoTH0smVStCmjAqiG+oYaSVKIJ4xEgzGlxN/eYDUZpKcWeGCQk56gkaU4yMlYL2DSc9NLmYVCudYsktuzPABXLierVTD3qZUgIZGp3iR7srccqJMJghrQPPTUw4QspQzMi40E41SRAeoB4JLBWIEx2OZieP4ZFVujCWypYwcKYuTowQ13rII9vJkenr395U/MsLUhOfhyMqktQQgeeL4pRBI+H0f9ilimDDhpYgrKi9FeI+Uggbm1LBhvD9Kfyf+JVyrezeVkv1yyyNPDgAh+AYeOAM1ME1aAAfYCDBI3gGL45xnpxX523emnOymX3wA877FyXikLU=</latexit><latexit sha1_base64="brUkHsmwbjExLPrfm3Bvs8RQTcI=">AAAB8XicdVDLSgNBEJyNrxhfUY9eBoPgKeyG+MhBCHrxZgTXBDZLmJ3MJkPmsczMCiEkf+HFg4pX/8abf+MkWSGKFjQUVd10d0UJo9q47qeTW1peWV3Lrxc2Nre2d4q7e/dapgoTH0smVStCmjAqiG+oYaSVKIJ4xEgzGlxN/eYDUZpKcWeGCQk56gkaU4yMlYL2DSc9NLmYVCudYsktuzPABXLierVTD3qZUgIZGp3iR7srccqJMJghrQPPTUw4QspQzMi40E41SRAeoB4JLBWIEx2OZieP4ZFVujCWypYwcKYuTowQ13rII9vJkenr395U/MsLUhOfhyMqktQQgeeL4pRBI+H0f9ilimDDhpYgrKi9FeI+Uggbm1LBhvD9Kfyf+JVyrezeVkv1yyyNPDgAh+AYeOAM1ME1aAAfYCDBI3gGL45xnpxX523emnOymX3wA877FyXikLU=</latexit><latexit sha1_base64="brUkHsmwbjExLPrfm3Bvs8RQTcI=">AAAB8XicdVDLSgNBEJyNrxhfUY9eBoPgKeyG+MhBCHrxZgTXBDZLmJ3MJkPmsczMCiEkf+HFg4pX/8abf+MkWSGKFjQUVd10d0UJo9q47qeTW1peWV3Lrxc2Nre2d4q7e/dapgoTH0smVStCmjAqiG+oYaSVKIJ4xEgzGlxN/eYDUZpKcWeGCQk56gkaU4yMlYL2DSc9NLmYVCudYsktuzPABXLierVTD3qZUgIZGp3iR7srccqJMJghrQPPTUw4QspQzMi40E41SRAeoB4JLBWIEx2OZieP4ZFVujCWypYwcKYuTowQ13rII9vJkenr395U/MsLUhOfhyMqktQQgeeL4pRBI+H0f9ilimDDhpYgrKi9FeI+Uggbm1LBhvD9Kfyf+JVyrezeVkv1yyyNPDgAh+AYeOAM1ME1aAAfYCDBI3gGL45xnpxX523emnOymX3wA877FyXikLU=</latexit>

W<latexit sha1_base64="rrzOMoTCq0SFtvo2WVqS0KsCtAc=">AAAB7nicdVBNS8NAEJ34WetX1aOXxSJ4Kon41VvRi8cKxhbaUCbbbbt0s4m7G6GE/gkvHlS8+nu8+W/cthGq6IOBx3szzMwLE8G1cd1PZ2FxaXlltbBWXN/Y3Nou7eze6ThVlPk0FrFqhqiZ4JL5hhvBmoliGIWCNcLh1cRvPDCleSxvzShhQYR9yXucorFSM2tTFKQx7pTKbsWdgsyRU9ernnnEy5Uy5Kh3Sh/tbkzTiElDBWrd8tzEBBkqw6lg42I71SxBOsQ+a1kqMWI6yKb3jsmhVbqkFytb0pCpOj+RYaT1KAptZ4RmoH97E/Evr5Wa3kWQcZmkhkk6W9RLBTExmTxPulwxasTIEqSK21sJHaBCamxERRvC96fkf+IfV6oV9+akXLvM0yjAPhzAEXhwDjW4hjr4QEHAIzzDi3PvPDmvztusdcHJZ/bgB5z3L0zLj64=</latexit><latexit sha1_base64="rrzOMoTCq0SFtvo2WVqS0KsCtAc=">AAAB7nicdVBNS8NAEJ34WetX1aOXxSJ4Kon41VvRi8cKxhbaUCbbbbt0s4m7G6GE/gkvHlS8+nu8+W/cthGq6IOBx3szzMwLE8G1cd1PZ2FxaXlltbBWXN/Y3Nou7eze6ThVlPk0FrFqhqiZ4JL5hhvBmoliGIWCNcLh1cRvPDCleSxvzShhQYR9yXucorFSM2tTFKQx7pTKbsWdgsyRU9ernnnEy5Uy5Kh3Sh/tbkzTiElDBWrd8tzEBBkqw6lg42I71SxBOsQ+a1kqMWI6yKb3jsmhVbqkFytb0pCpOj+RYaT1KAptZ4RmoH97E/Evr5Wa3kWQcZmkhkk6W9RLBTExmTxPulwxasTIEqSK21sJHaBCamxERRvC96fkf+IfV6oV9+akXLvM0yjAPhzAEXhwDjW4hjr4QEHAIzzDi3PvPDmvztusdcHJZ/bgB5z3L0zLj64=</latexit><latexit sha1_base64="rrzOMoTCq0SFtvo2WVqS0KsCtAc=">AAAB7nicdVBNS8NAEJ34WetX1aOXxSJ4Kon41VvRi8cKxhbaUCbbbbt0s4m7G6GE/gkvHlS8+nu8+W/cthGq6IOBx3szzMwLE8G1cd1PZ2FxaXlltbBWXN/Y3Nou7eze6ThVlPk0FrFqhqiZ4JL5hhvBmoliGIWCNcLh1cRvPDCleSxvzShhQYR9yXucorFSM2tTFKQx7pTKbsWdgsyRU9ernnnEy5Uy5Kh3Sh/tbkzTiElDBWrd8tzEBBkqw6lg42I71SxBOsQ+a1kqMWI6yKb3jsmhVbqkFytb0pCpOj+RYaT1KAptZ4RmoH97E/Evr5Wa3kWQcZmkhkk6W9RLBTExmTxPulwxasTIEqSK21sJHaBCamxERRvC96fkf+IfV6oV9+akXLvM0yjAPhzAEXhwDjW4hjr4QEHAIzzDi3PvPDmvztusdcHJZ/bgB5z3L0zLj64=</latexit>

4

2

1

0

3

3 27 3 9

0.07

1.29

0.64

2.21

2.79

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