JMAK の式の一般化と粒子サイズ分布の計算 by T.Koyama

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1 MAK by T.Koyama

2 MAK MAK f () = exp{ fex () = exp (') v(, ') ' () (') ' v (, ') ' f (), (), v (, ') f () () f () () v (, ') f () () v (, ') f () () () = + {exp( A) () f () = exp( K ) () K,,, A *** *************************************************************************** N () () N () dn( ) () = f ( ) () f ( ) ************************************************************************************ v (, ') v (, ') = v ( ') ( ') f() = exp (') v(, ') ' = exp [ + { exp( A')] v( ') ' = exp v ( ') ' { v ( ') v ( ') exp( A') ' + = exp v ( ') ' + v ( ') exp( A ') '

3 " = ' " = ', "( ' = ) =, "( ' = ) = ' = " ' = " f + v v A ( ) = exp ( ") " ( ") exp{ ( ") " + = + + exp v( ") " exp( A) v( ") exp( A ") " + exp v( ") " exp( A) v( ") exp( A ") " f ( ) + = + v( ") " + exp( A) v( ") exp( A ") " = l{ f ( ) + v ( ") " exp( A) v ( ") exp( A") " l{ f( ) l = = f ( ) *** *************************************************************************** f ( ) = exp ( ') v(, ') ' = exp ( ') v( ') ' " = ', "( ' = ) =, "( ' = ) = " = ' ' = ", ' = " f () = exp (') v( ') ' exp ( ") v(")( ") = = exp ( ") v( ") " ( ") v( " ) ( " ) " exp( α) () ************************************************************************************

4 g( f ) l f ( (4) ) + g( f) l = v ( ") " exp( A) v ( ") exp( A") " f( ) dg + A v ( ) exp( A) v ( ") exp( A") " exp( Av ) ( ) exp( A) = + + A v ( ) exp( A) v ( ") exp( A") " v ( ) = + = A v ( ) + exp( A) v ( ") exp( A") " dv A A d g = exp( A) v( ") exp( A ") " + exp( A) v( ) exp( A) dv A A = exp( A) v( ") exp( A ") " + v( ) dv dg A dv dg + = A + A v () + v () = A + A v () + A A v() d g dv dg = + dv() + d g dg + A v = + A () (5) v () = d g dv dg + d g dg = A + A v() = A + A v() v d g dg = + () A (6) ()(4) g( f ) l = l l{ exp( ) = K = K f( ) [ exp( K )] dg = K d g = ( ) K (7)

5 dv() + av() = () v() = () = ((7)) " ( ) { { v( ) = ( ") exp a ' " + C exp a ' + C exp a ' β α α α = ( ") exp a " + C " + C exp ( a + C) + Cexp ( a + C) β ( ) { { = ( ") exp ( a " + C) " + ( ") exp ( a " + C) " + C + C exp { ( a C ) + β = (")exp( a" + C ) " + Cβ + C + C exp ( a + C) = (")exp a" + C "exp ( a+ C) ( ) { = exp( C) ( ") exp( a") " exp( a) exp( C) = exp( a) ( ') exp( a ') ' { v() = () = v() = ( Cβ + C + C) exp( C) =, Cβ + C + C = (5)(7) + d g dg + v () = exp A A exp A ' ' = exp A { K( ) ( ') AK ( ') exp A ' ' + v () + K AK + = exp A ( ) ( ') + ( ') exp ' ' A (8) v (), :[m] A,, :[s K :[s ] ] 4

6 v () [m] [m ] A : = [] + [s ] [s ] [s] [] = = [s ] K [s ] ( ) (') ' [s ][] [s ] [] s + + = = = [s ] AK [s ][s ] ( ) (') ' [s ][] [s ] [] s + + = = = [s ] (8)(8) v () ( ) / = dn = + ( ) + ( ) { f ( )/ + [{ f ( ) / / ( )] =,,,,, + ( ) = + {exp( A) f ( ) + ( ) = { f( ) ( ) (9) ( ) = ( ) = dn/ = { f( ) ( ) { f ( )/ { f ( )/ + 5

7 f (, ) c fc (, ) = exp ( ') v (, ') ' () { f ( )/ { f( ) = f (, ) f (, + ) c c f( ) fc(, ) fc(, + ) fc(, + ) fc(, ) = = f( ) df (, ) c d lim = = exp ( ') v(, ') ' d = (')(,') v ' exp ( ') v(, ') ' = ( ) v(, )exp ( ') v(, ') ' () + fc(, ) > fc(, + ) + [{ f ( ) / / ( )] df (, ) c ( ) ( ) v (, )exp ( ') v (, ') ' ( ) f( )/ ( ) = = ( ) { f( ) ( )( ) v (, ) v (, ') = ( ) exp ( ') ' f () v (, ')/ (8) f () () f ( )/ ( ) ( ) + = + = (8) v () K AK K AK K = ( ) ( ') + ( ') ' = + = ( + A ) () 6

8 v (, ') K = + A { ( ') ( ') ( ) = + { exp( A) = + exp( A) = exp( A) = exp( A) (4) f( )/ v(, ) v (, ') exp ( ') ' ( ) = f( ) K K (5) = { ( ) + A( ) exp exp( A') { ( ') + A( ') ' f( ) K = { ( ) + A( ) exp K { ( f( ) { ') ( ') A exp( A ') ' + f( )/ K { ( ) A( ) ( ) = + f( ) { K A A exp { ( ') + ( ') exp( ') ' (6) ' = s, ' = s = ' =, ' = s = = ' = s ' = ds' f( )/ K { ( ) A( ) ( ) = + f( ) { K A A exp { ( ') + ( ') exp( ') ' K = { ( ) + A( ) f( ) exp { K ( s As ) exp{ A( s)( ds) + K = { ( ) + A( ) f( ) { K A s As As ds exp exp( ) ( + ) exp( ) K = { ( ) + A( ) f( ) { A K A A exp exp( ) ( + ) exp( ) (7) 7

9 ( f + 4f + f ) ( f + 4 f + f ) + ( f + 4 f + f 4) + ( f f 5 + f 6) + ( f + 4 f + f ) = ( f + 4f + f + 4f + f4 + 4f5 + f6 + + f + 4f + f ) = [, ] i i = i = i = f () / F ( i ) i F( i) = { f( i /) + 4 f( i) + f( i + /) = { f( i /) + 4 f( i) + f( i + /) 6 i = F( ) = {4 f( ) + f( + /) F ( i ) 6 = = (7) = = ma i x 8

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