Date: Fri, 1 Oct 2004 12:46:36 +0100 (GMT/BST) ANISOTROPIC STATISTICAL MECHANICS I constructed a Theory of Anisotropic Statistical Mechanics for spherical particles and anisotropic fields. I proved this field equations. MY AXIOMS FOR ANISOTROPIC SPACE TIMES : DIMENSIONAL DESCRIPTION : Anisotropic space has integer, fractional, irrational dimensions. SPACIAL DESCRIPTION : RATIONAL SPACE : It has integer dimensions. FRACTIONAL SPACE : It has fractional dimensions. IRRATIONAL SPACE : It has irrational dimensions. PHASE SPACES RATIONAL SPACE : It's on 3 dimensional rational space, phase space is 6 dimensional rational space. FRACTIONAL SPACE : It's on 4 dimensional fractional space, phase space's 8 dimensional fractional space. IRRATIONAL SPACE : It's on 4 dimensional irrational space, phase space's 8 dimensional irrational space. QUANTITY SPACES RATIONAL SPACE : It's on 3 dimensional rational space FRACTIONAL SPACE : It's on 4 dimensional fractional space. IRRATIONAL SPACE : It's on 4 dimensional irrational space. PHASE SPACES : RATIONAL SPACE : nr(p,x)1 = 4.pi.P1^3.V1 / 3.h^3 nr(p,x)2 = 4.pi.P2^3.V2 / 3.h^3 nr(p,x)3 = 4.pi.P3^3.V3 / 3.h^3 v1,V2,V3 are the sizes of three dimensional space. FRACTIONAL SPACE : nf(p,x)1 = pi.P1^4.F1 / 3.h^4 nf(p,x)2 = pi.P2^4.F2 / 3.h^4 nf(p,x)3 = pi.P3^4.F3 / 3.h^4 nf(p,x)4 = pi.P4^4.F4 / 3.h^4 F1,F2,F3,F4 are the sizes of four dimensional space. IRRATIONAL SPACE : ni(p,x)1 = pi.P1^4.R1 / 3.h^4 ni(p,x)2 = pi.P2^4.R2 / 3.h^4 ni(p,x)3 = pi.P3^4.R3 / 3.h^4 ni(p,x)4 = pi.P4^4.R4 / 3.h^4 R1,R2,R3,R4 are the sizes of four dimensional space QUANTITY SPACE : RATINAL SPACE : k1 is a constant in rational space for distributing of the spherical particles.It is Boltzmann's constant. Er = 3.k1.T / 2 = h.f1 E1 = 3.k1.T1 / 2 = h.f11 E2 = 3.k1.T2 / 2 = h.f12 E3 = 3.k1.T3 / 2 = h.f13 FRACTIONAL SPACE : k2 is a constant in fractional space for distributing of the spherical particles. Er = 4.k2.T / 2 = h.f2 E1 = 4.k2.T1 / 2 = h.f21 E2 = 4.k2.T2 / 2 = h.f22 E3 = 4.k2.T3 / 2 = h.f23 E4 = 4.k2.T4 / 2 = h.f24 IRRATIONAL SPACE : k3 is a constant in irrational space for distributing of the spherical particles. Ei = 4.k3.T / 2 = h.f3 E1 = 4.k3.T1 / 2 = h.f31 E2 = 4.k3.T2 / 2 = h.f32 E3 = 4.k3.T3 / 2 = h.f33 E4 = 4.k3.T4 / 2 = h.f34 EQUATION OF ANISOTROPIC ENERGY FOR SPHERICAL PARTICLES: E = Er + Ef + Ei = 3.k1.T / 2 + 4.k2.T / 2 + 4.k3.T / 2 = h.f1 + h.f2 + hf3 ANISOTROPIC DISTRIBUTING EQUATIONS FOR SPHERICAL PARTICLES MAXWELL BOLTZMANN FUNCTIONS RATIONAL SPACE nr = g / e^( Er / k1.T ) n1 = g1 / e^( E1 / k1.T1 ) n2 = g2 / e^( E2 / k1.T2 ) n3 = g3 / e^( E3 / k1.T3 ) FRACTIONAL SPACE nf = g / e^( Ef / k2.T ) n1 = g1 / e^( E1 / k2.T1 ) n2 = g2 / e^( E2 / k2.T2 ) n3 = g3 / e^( E3 / k2.T3 ) n4 = g4 / e^( E4 / k2.T4 ) IRRATIONAL SPACE ni = g / e^( Ei / k3.T ) n1 = g1 / e^( E1 / k3.T1 ) n2 = g2 / e^( E2 / k3.T2 ) n3 = g3 / e^( E3 / k3.T3 ) n4 = g4 / e^( E4 / k3.T4 ) FERMI DIRAC DISTRIBUTION RATIONAL SPACE nr = g / e^( Er / k1.T )+ 1 n1 = g1 / e^( E1 / k1.T1 ) + 1 n2 = g2 / e^( E2 / k1.T2 ) + 1 n3 = g3 / e^( E3 / k1.T3 ) + 1 FRACTIONAL SPACE nf = g / e^( Ef / k2.T ) + 1 n1 = g1 / e^( E1 / k2.T1 ) + 1 n2 = g2 / e^( E2 / k2.T2 ) + 1 n3 = g3 / e^( E3 / k2.T3 ) + 1 n4 = g4 / e^( E4 / k2.T4 ) + 1 IRRATIONAL SPACE ni = g / e^( Ei / k3.T ) + 1 n1 = g1 / e^( E1 / k3.T1 ) + 1 n2 = g2 / e^( E2 / k3.T2 ) + 1 n3 = g3 / e^( E3 / k3.T3 ) + 1 n4 = g4 / e^( E4 / k3.T4 ) + 1 BOSE EINSTEIN DISTRIBUTION RATIONAL SPACE nr = g / e^( Er / k1.T )- 1 n1 = g1 / e^( E1 / k1.T1 ) - 1 n2 = g2 / e^( E2 / k1.T2 ) - 1 n3 = g3 / e^( E3 / k1.T3 ) - 1 FRACTIONAL SPACE nf = g / e^( Ef / k2.T ) - 1 n1 = g1 / e^( E1 / k2.T1 ) - 1 n2 = g2 / e^( E2 / k2.T2 ) - 1 n3 = g3 / e^( E3 / k2.T3 ) - 1 n4 = g4 / e^( E4 / k2.T4 ) - 1 IRRATIONAL SPACE ni = g / e^( Ei / k3.T ) - 1 n1 = g1 / e^( E1 / k3.T1 ) - 1 n2 = g2 / e^( E2 / k3.T2 ) - 1 n3 = g3 / e^( E3 / k3.T3 ) - 1 n4 = g3 / e^( E4 / k3.T4 ) - 1 We can explain on the deviations of known isotropic statistical mechanics by anisotropic statistical mechanics. Sincerely Ozan Hasimi OKTAR ********************************* Date: Fri, 1 Oct 2004 12:58:22 +0100 (GMT/BST) ANISOTROPIC STATISTICAL MECHANICS OF HYPERBOLICAL PARTICLES I constructed a theory of anisotropic statistical mechanics for hyperbolical particles. I proved this field equations. MY AXIOMS FOR ANISOTROPIC SPACE TIMES : DIMENSIONAL DESCRIPTION : Anisotropic space has integer, fractional, irrational dimensions. SPACIAL DESCRIPTION : RATIONAL SPACE : It has integer dimensions. FRACTIONAL SPACE : It has fractional dimensions. IRRATIONAL SPACE : It has irrational dimensions. PHASE SPACES RATIONAL SPACE : It's on 3 dimensional rational space, phase space is 6 dimensional rational space. FRACTIONAL SPACE : It's on 4 dimensional fractional space, phase space's 8 dimensional fractional space. IRRATIONAL SPACE : It's on 4 dimensional irrational space, phase space's 8 dimensional irrational space. QUANTITY SPACES RATIONAL SPACE : It's on 3 dimensional rational space FRACTIONAL SPACE : It's on 4 dimensional fractional space. IRRATIONAL SPACE : It's on 4 dimensional irrational space. PHASE SPACES : RATIONAL SPACE : nr(p,x) = 6.pi.sinh^3(p/3).6.pi.sinh^3(x/3) / h^3 FRACTIONAL SPACE : nf(p,x) = 8.pi.sinh^4(p/4).8.pi.sinh^4(x/4) / h^4 IRRATIONAL SPACE : ni(p,x) = 8.pi.sinh^4(p/4).8.pi.sinh^4(x/4) / h^4 QUANTITY SPACE : RATINAL SPACE : k1 is a constant in rational space for distributing of the particles. Er = 3.k1.T / 2 = h.f1 E1 = 3.k1.T1 / 2 = h.f11 E2 = 3.k1.T2 / 2 = h.f12 E3 = 3.k1.T3 / 2 = h.f13 FRACTIONAL SPACE : k2 is a constant in fractional space for distributing of the particles. Er = 4.k2.T / 2 = h.f2 E1 = 4.k2.T1 / 2 = h.f21 E2 = 4.k2.T2 / 2 = h.f22 E3 = 4.k2.T3 / 2 = h.f23 IRRATIONAL SPACE : k3 is a constant in irrational space for distributing of the particles. Ei = 4.k3.T / 2 = h.f3 E1 = 4.k3.T1 / 2 = h.f31 E2 = 4.k3.T2 / 2 = h.f32 E3 = 4.k3.T3 / 2 = h.f33 EQUATION OF ANISOTROPIC ENERGY : E = Er + Ef + Ei = 3.k1.T / 2 + 4.k2.T / 2 + 4.k3.T / 2 = h.f1 + h.f2 + hf3 ANISOTROPIC DISTRIBUTING EQUATIONS MAXWELL BOLTZMANN FUNCTIONS RATIONAL SPACE nr = g / e^( Er / k1.T ) n1 = g1 / e^( E1 / k1.T1 ) n2 = g2 / e^( E2 / k1.T2 ) n3 = g3 / e^( E3 / k1.T3 ) FRACTIONAL SPACE nf = g / e^( Ef / k2.T ) n1 = g1 / e^( E1 / k2.T1 ) n2 = g2 / e^( E2 / k2.T2 ) n3 = g3 / e^( E3 / k2.T3 ) IRRATIONAL SPACE ni = g / e^( Ei / k3.T ) n1 = g1 / e^( E1 / k3.T1 ) n2 = g2 / e^( E2 / k3.T2 ) n3 = g3 / e^( E3 / k3.T3 ) FERMI DIRAC DISTRIBUTION RATIONAL SPACE nr = g / e^( Er / k1.T )+ 1 n1 = g1 / e^( E1 / k1.T1 ) + 1 n2 = g2 / e^( E2 / k1.T2 ) + 1 n3 = g3 / e^( E3 / k1.T3 ) + 1 FRACTIONAL SPACE nf = g / e^( Ef / k2.T ) + 1 n1 = g1 / e^( E1 / k2.T1 ) + 1 n2 = g2 / e^( E2 / k2.T2 ) + 1 n3 = g3 / e^( E3 / k2.T3 ) + 1 IRRATIONAL SPACE ni = g / e^( Ei / k3.T ) + 1 n1 = g1 / e^( E1 / k3.T1 ) + 1 n2 = g2 / e^( E2 / k3.T2 ) + 1 n3 = g3 / e^( E3 / k3.T3 ) + 1 BOSE EINSTEIN DISTRIBUTION RATIONAL SPACE nr = g / e^( Er / k1.T )- 1 n1 = g1 / e^( E1 / k1.T1 ) - 1 n2 = g2 / e^( E2 / k1.T2 ) - 1 n3 = g3 / e^( E3 / k1.T3 ) - 1 FRACTIONAL SPACE nf = g / e^( Ef / k2.T ) - 1 n1 = g1 / e^( E1 / k2.T1 ) - 1 n2 = g2 / e^( E2 / k2.T2 ) - 1 n3 = g3 / e^( E3 / k2.T3 ) - 1 IRRATIONAL SPACE ni = g / e^( Ei / k3.T ) - 1 n1 = g1 / e^( E1 / k3.T1 ) - 1 n2 = g2 / e^( E2 / k3.T2 ) - 1 n3 = g3 / e^( E3 / k3.T3 ) - 1 Sincerely Ozan Hasimi OKTAR ********************************* Date: Fri, 1 Oct 2004 13:18:47 +0100 (GMT/BST) STATISTICAL MECHANICS FOR HYPERBOLICAL PARTICLES When I was studying on Hyperbolical Spaces, I constructed a Theory of Statistical Mechanics for Hyperbolical Particles. I proved this field equations. They are isotropic spaces. MY AXIOMS FOR ISOTROPIC SPACE TIMES : DIMENSIONAL DESCRIPTION : Isotropic space has integer dimensions. SPACIAL DESCRIPTION : RATIONAL SPACE : It has integer dimensions. PHASE SPACES RATIONAL SPACE : It's on 3 dimensional rational space, phase space is 6 dimensional rational space. QUANTITY SPACES RATIONAL SPACE : It's on 3 dimensional rational space PHASE SPACES : RATIONAL SPACE : nr(p,x) = 6.pi.sinh^3(p/3).6.pi.sinh^3(x/3) / h^3 QUANTITY SPACE : RATIONAL SPACE : k1 is a constant in rational space for distributing of the particles. Er = 3.k1.T / 2 = h.f1 EQUATION OF ISOTROPIC ENERGY : E = Er = 3.k1.T / 2 = h.f1 ISOTROPIC DISTRIBUTING EQUATIONS MAXWELL BOLTZMANN FUNCTIONS RATIONAL SPACE nr = g / e^( Er / k1.T ) FERMI DIRAC DISTRIBUTION RATIONAL SPACE nr = g / e^( Er / k1.T )+ 1 BOSE EINSTEIN DISTRIBUTION RATIONAL SPACE nr = g / e^( Er / k1.T )- 1 Sincerely Ozan Hasimi OKTAR Date: Fri, 3 Dec 2004 12:17:29 GMT Dear Mr. Ray dickenson ; 5 DIMENSIONAL DIFFERANTIAL OPERATOR SPACES 3 DIMENSIONAL OPERATOR SPACES They are gradient,divergence,rotational operators. You know that them. 5 DIMENSIONAL OPERATOR SPACES 5 DIMENSIONAL DEL OPERATOR Del = [@ /@x1] e1 + [ @ / @x2] e2 + [@ / @x3] e3 + [ @ / @x4 ]e4 + [ @ / @x5 ] e5 5 DIMENSIONAL GRADIENT OPERATOR grad Q = [@Q1 /@x1] e1 + [@Q2 / @x2] e2 + [@Q3 / @x3] e3 + [ @Q4 / @x4 ]e4 + [ @Q5 / @x5 ] e5 5 DIMENSIONAL DIVERGENCE OPERATOR div Q = [@Q1 /@x1] + [@Q2 / @x2] + [@Q3 / @x3] + [ @Q4 / @x4 ] + [ @Q5 / @x5 ] 5 DIMENSIONAL ROTATINOAL OPERATOR rot Q = + [@Q2 /@x4] e1 + [@Q3/@x5]e1 - [@Q4 /@x2] e1 - [@Q5/@x3]e1 + [@Q1 / @x3]e2 + [@Q5 /@x4]e2 - [@Q3 / @x1 ]e2 - [@Q4 /@x5] e2 + [@Q4 / @x1]e3 + [@Q5 /@x2] e3 - [@Q1 / @x4]e3 - [@Q2 /@x5] e3 + [@Q3 / @x2]e4 + [@Q1 /@x5] e4 - [@Q2 / @x3]e4 - [@Q5 /@x1] e4 + [@Q2 / @x1]e5 + [@Q4 /@x3] e5 - [@Q1 / @x2]e5 - [@Q3 /@x4] e5 We can use for every 5 dimensional spaces. Sincerely Ozan Hasimi OKTAR javaquant@lycos.com