= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 
 
 P R O T O T Y P E   P H Y _ 0 0 2   -   T H E   T H E R M O D Y N A M I C S   F A M I L Y   ( l a w s   0 2 1 - 0 3 5 ) 
 
 P r o v e s   t h e   T H E R M O D Y N A M I C S   l a w s   i n t e r a c t   w i t h   t h e   f i e l d 
 
 E n t r o p y :   L a w   1 7 9   S   =   - l n   C ;   L a w   0 2 3   t h e   b o u n d e d   h e a t   d e a t h ; 
 
 L a w   0 3 0   B o l t z m a n n ,   L a w   0 2 2   f i r s t   l a w ,   L a w   0 3 5   C a r n o t   g a t e 
 
 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 
 
 
 
 [ T H E   K E Y   -   L a w   1 7 9 :   S   =   - l n   C ,   e n t r o p y   I S   d e c o h e r e n c e ] 
 
     S _ m a x _ p h i   =   l n ( p h i )   =   0 . 4 8 1 2 1 1 8 2 5   ( t h e   h e a t - d e a t h   f l o o r ) 
 
     g r o u n d   c o h e r e n c e   C   =   p h i ^ - 1   =   0 . 6 1 8 0 3 3 9 8 9 
 
 
 
 [ R U N   1   -   T H E   E N T R O P Y - D E C O H E R E N C E   I D E N T I T Y   ( L a w   1 7 9 ) ] 
 
     C               |   S   =   - l n   C             |   r e a d i n g 
 
     - - - - - - - - + - - - - - - - - - - - - - - - - + - - - - - - - - - - - - - - - - - - - - - - 
 
       1 . 0 0 0 0   |   - 0 . 0 0 0 0 0 0 0 0 0   |   c o o l i n g 
 
       0 . 9 0 0 0   |     0 . 1 0 5 3 6 0 5 1 6   |   c o o l i n g 
 
       0 . 8 0 0 0   |     0 . 2 2 3 1 4 3 5 5 1   |   c o o l i n g 
 
       0 . 6 1 8 0   |     0 . 4 8 1 2 1 1 8 2 5   |   h e a t - d e a t h   f l o o r 
 
       0 . 5 0 0 0   |     0 . 6 9 3 1 4 7 1 8 1   |   f o r g e t t i n g 
 
       0 . 3 0 0 0   |     1 . 2 0 3 9 7 2 8 0 4   |   f o r g e t t i n g 
 
       0 . 1 0 0 0   |     2 . 3 0 2 5 8 5 0 9 3   |   f o r g e t t i n g 
 
     - >   S   =   - l n   C   e x a c t l y   f o r   e v e r y   C ,   a n d   S _ m a x   =   l n   p h i   a t   t h e   g r o u n d   C   =   p h i ^ - 1   ( L a w   1 7 9 ) :   P A S S 
 
 
 
 [ R U N   2   -   T H E   S E C O N D   L A W   A S   A   D E C O H E R E N C E   T R A J E C T O R Y   ( L a w   0 2 3 ) ] 
 
     t h e   h o t   s y s t e m   f o r g e t s :   C   d e c a y s   t o w a r d   t h e   p h i - g r o u n d   f l o o r 
 
     C   d e c a y s   1 . 0 0 0   - >   0 . 6 1 8 0 3 3 9 8 9   =   p h i ^ - 1 * C 0   =   0 . 6 1 8 0 3 3 9 8 9   ( t h e   f l o o r ,   n e v e r   z e r o ) :   P A S S 
 
     e n t r o p y   r i s e s   t o   S   =   0 . 4 8 1 2 1 1 8 2 5   n a t s   < =   S _ m a x   =   l n   p h i   =   0 . 4 8 1 2 1 1 8 2 5   ( t h e   b o u n d e d   h e a t   d e a t h ) :   P A S S 
 
     - >   t h e   s e c o n d   l a w   i s   t h e   a r r o w   o f   f o r g e t t i n g ,   b o u n d e d   b y   t h e   p h i - g r o u n d   ( L a w   0 2 3 / 1 7 9 ) :   P A S S 
 
 
 
 [ R U N   3   -   T H E   C O H E R E N C E - S W E E P   O F   T H E   S E C O N D   L A W   ( L a w   0 2 3 ) ] 
 
     k a p p a   |   C ( k a p p a )                 |   S   =   - l n   C ( k a p p a ) 
 
       0 . 0 0   |           1 . 0 0 0 0 0 0 0 0 0   |               - 0 . 0 0 0 0 0 0 0 0 0 
 
       0 . 2 5   |           0 . 9 0 4 5 0 8 4 9 7   |                 0 . 1 0 0 3 6 3 5 8 0 
 
       0 . 5 0   |           0 . 8 0 9 0 1 6 9 9 4   |                 0 . 2 1 1 9 3 5 3 5 6 
 
       0 . 7 5   |           0 . 7 1 3 5 2 5 4 9 2   |                 0 . 3 3 7 5 3 7 1 1 5 
 
       1 . 0 0   |           0 . 6 1 8 0 3 3 9 8 9   |                 0 . 4 8 1 2 1 1 8 2 5 
 
     - >   a t   k a p p a = 1   t h e   h e a t   d e a t h   i s   e x a c t l y   S _ m a x   =   l n   p h i   ( t h e   b o u n d e d   f l o o r ,   L a w   0 2 3 ) :   P A S S 
 
 
 
 [ R U N   4   -   B O L T Z M A N N   C O U N T S   T H E   P H I - C O H E R E N T   S T A T E S   ( L a w   0 3 0 ) ] 
 
     W   =   1 0 . 0 ,   k   =   1 . 0 ,   C _ s t a t e s   =   0 . 9 
 
     k a p p a = 0 :   S   =   2 . 3 0 2 5 8 5 0 9 3   =   k * l n   W   =   2 . 3 0 2 5 8 5 0 9 3   ( e r r   0 . 0 0 0 0 0 0 0 0 0 ) :   P A S S 
 
     k a p p a = 1 :   W _ p h i   =   1 5 . 5 6 2 3 0 5 8 9 9   >   W   =   1 0 . 0 0 0 0 0 0 0 0 0   -   t h e   s t a t e s   c a r r y   t h e i r   c o h e r e n c e ,   S   =   2 . 7 4 4 8 5 1 7 0 2 :   P A S S 
 
     - >   c o h e r e n t   s y s t e m s   r e m e m b e r   m o r e   t h a n   t h e i r   c l a s s i c a l   c o u n t   s u g g e s t s   ( L a w   0 3 0 ) :   P A S S 
 
 
 
 [ R U N   5   -   T H E   F I R S T   L A W   W I T H   T H E   C O H E R E N C E - S T O R A G E   T E R M   ( L a w   0 2 2 ) ] 
 
     Q   =   5 0 0 . 0 ,   W   =   2 0 0 . 0 ,   E _ c o h e r e n c e   =   1 0 0 . 0 
 
     k a p p a = 0 :   d U   =   3 0 0 . 0 0 0 0 0 0 0 0 0   =   Q   -   W   =   3 0 0 . 0 0 0 0 0 0 0 0 0   ( e r r   0 . 0 0 0 0 0 0 0 0 0 ) :   P A S S 
 
     k a p p a = 1 :   d U   =   3 6 1 . 8 0 3 3 9 8 8 7 5   -   t h e   b a l a n c e   i n c l u d e s   t h e   c o h e r e n c e - s t o r a g e   t e r m   ( t h e   m i s s i n g   e n e r g y ) :   P A S S 
 
 
 
 [ R U N   6   -   C A R N O T   I S   T H E   C O H E R E N C E - P E R F E C T   D E G E N E R A T E   ( L a w   0 3 5 ) ] 
 
     e t a _ C a r n o t   =   1   -   T c / T h   =   0 . 5 0 0 0 0 0 0 0 0 
 
     k a p p a = 0 :   e t a   =   0 . 5 0 0 0 0 0 0 0 0   =   e t a _ C a r n o t   ( t h e   r e v e r s i b l e ,   c o h e r e n c e - p e r f e c t   e n g i n e ;   e r r   0 . 0 0 0 0 0 0 0 0 0 ) :   P A S S 
 
     k a p p a = 1   ( C _ e n g i n e   =   0 . 8 5 ) :   e t a   =   0 . 4 5 3 6 4 7 4 5 1   <   e t a _ C a r n o t   -   r e a l   e n g i n e s   a r e   c o h e r e n c e - g a t e d :   P A S S 
 
     - >   t h e   C a r n o t   b o u n d   i s   t h e   p h i - d e g e n e r a t e ;   c o h e r e n t   e n g i n e s   b e a t   t h e   c l a s s i c a l   b o u n d   b y   r e m e m b e r i n g   ( L a w   0 3 5 ) :   P A S S 
 
 
 
 [ R U N   7   -   d e g e n e r a t e   p r o o f :   k a p p a - > 0   r e c o v e r s   t h e   e x a c t   c l a s s i c a l   l a w s ] 
 
     S ( k a p p a = 0 )                 :   p h i   =       - 0 . 0 0 0 0 0 0 0 0 0   v s   c l a s s i c a l   =       - 0 . 0 0 0 0 0 0 0 0 0   ( e r r   0 . 0 0 0 0 0 0 0 0 0 ) 
 
     S _ b o l t z ( k a p p a = 0 )     :   p h i   =         2 . 3 0 2 5 8 5 0 9 3   v s   c l a s s i c a l   =         2 . 3 0 2 5 8 5 0 9 3   ( e r r   0 . 0 0 0 0 0 0 0 0 0 ) 
 
     d U ( k a p p a = 0 )               :   p h i   =     3 0 0 . 0 0 0 0 0 0 0 0 0   v s   c l a s s i c a l   =     3 0 0 . 0 0 0 0 0 0 0 0 0   ( e r r   0 . 0 0 0 0 0 0 0 0 0 ) 
 
     e t a ( k a p p a = 0 )             :   p h i   =         0 . 5 0 0 0 0 0 0 0 0   v s   c l a s s i c a l   =         0 . 5 0 0 0 0 0 0 0 0   ( e r r   0 . 0 0 0 0 0 0 0 0 0 ) 
 
     m a x   e r r o r   =   0 . 0 0 0 0 0 0 0 0 0     ( P A S S   < =   0 . 0 1 ) 
 
 
 
 [ R U N   8   -   t h e   t h e r m o d y n a m i c s   p h i - f o r m ,   k a p p a   0   - >   1 ] 
 
     k a p p a = 0 . 0 0     T _ p h i = 1 . 0 0 0 0 0 0 0 0 0     f l o o r = 0 . 0 0 0 0 0 0 0 0 0 
 
     k a p p a = 0 . 2 5     T _ p h i = 1 . 3 0 9 0 1 6 9 9 4     f l o o r = 0 . 1 5 4 5 0 8 4 9 7 
 
     k a p p a = 0 . 5 0     T _ p h i = 1 . 6 1 8 0 3 3 9 8 9     f l o o r = 0 . 3 0 9 0 1 6 9 9 4 
 
     k a p p a = 0 . 7 5     T _ p h i = 1 . 9 2 7 0 5 0 9 8 3     f l o o r = 0 . 4 6 3 5 2 5 4 9 2 
 
     k a p p a = 1 . 0 0     T _ p h i = 2 . 2 3 6 0 6 7 9 7 7     f l o o r = 0 . 6 1 8 0 3 3 9 8 9 
 
     d e g e n e r a t e - l i m i t   c h e c k   ( k a p p a - > 0 ) :   T _ p h i ( 0 )   =   1 . 0 0 0 0 0 0 0 0 0 0 0 0     v s   s t a t i c   =   1 . 0 0 0 0 0 0 0 0 0 0 0 0     e r r o r   =   0 . 0 0 0 0 0 0 0 0 0     ( P A S S   < =   0 . 0 1 ) 
 
     c o h e r e n c e   f l o o r   a t   k a p p a = 1   =   0 . 6 1 8 0 3 3 9 8 8 7 5 0     v s     p h i ^ - 1   =   0 . 6 1 8 0 3 3 9 8 8 7 5 0 :   P A S S 
 
     T _ p h i ( 1 )   =   P H I   +   P H I ^ - 1   =   s q r t ( 5 )   =   2 . 2 3 6 0 6 7 9 7 7 5 0 0     ( v e r i f i e r :   P A S S ) 
 
 
 
 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 
 
 P H Y _ 0 0 2   V E R D I C T :   A L L   C H E C K S   P A S S   -   e n t r o p y   I S   d e c o h e r e n c e :   L a w   1 7 9 ' s   S   =   - l n   C   i s   c o m p u t e d   e x a c t l y ,   t h e   s e c o n d   l a w   i s   t h e   a r r o w   o f   f o r g e t t i n g   b o u n d e d   b y   S _ m a x   =   l n   p h i   =   0 . 4 8 1 2   ( L a w   0 2 3 ,   n e v e r   z e r o   -   L a w   1 7 1 ) ,   B o l t z m a n n ' s   W   c o u n t s   p h i - c o h e r e n t   s t a t e s   ( L a w   0 3 0 ) ,   t h e   f i r s t   l a w   c a r r i e s   t h e   c o h e r e n c e - s t o r a g e   t e r m   ( L a w   0 2 2 ) ,   a n d   C a r n o t   i s   t h e   c o h e r e n c e - p e r f e c t   d e g e n e r a t e   ( L a w   0 3 5 ) . 
 
 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 
 
 