Configurations Mathematically describe a configuration Specify coordinates of eachatom with=14=1/12 73 Specify a slot for each atom ith 1.4R ··2 121923
Configurations • Mathematically describe a configuration – Specify coordinates of each “atom” Ø1 3 4 5 ø Ø ø Ø ø Ø ø { } with i=1,..,4 r 1 = Œ œ , r = Œ œ , r = Œ œ , r = Œ œ ri 1 2 º ß º ß 4 4 2 3 3 º ß ºß – Specify a slot for each atom {ri} with i=1,..,4 R = [1 12 19 23]
Energy of a state Each state(configuration) has an energy level associated with it H(q)=∑9P-L(99 Hamiltonian H=T+y=e tot Energy of configuration Objective function value of design kinetic potential Energy
� � Energy of a state • Each state (configuration) has an energy level associated with it ( , , � H q q t ) = �qii p - L (qqt , , ) i Hamiltonian H =+= T V Etot Energy of configuration = Objective function value of design kinetic potential Energy
Energy sample problem Define energy function for "atom" sample problem E;=mgy, t ∑ x i -xi+ly potential/≠ energy kinetic energy E()=∑E() Absolute and relative position of each atom contributes to Energy
Energy sample problem • Define energy function for “atom” sample problem 1 � i +�(( xi - xj) + ( y - y Ei = mgy j) ) N 2 2 2 i „ potential ���� j i ��������� energy kinetic energy N E (R ) = � E r ) Absolute and relative position of i ( i i=1 each atom contributes to Energy
Compute Energy of Config. A Energy of initial configuration E1=1.10.1+√5+18+20=20.95 E,=110.2+√5+√5+5=2671 E1=110.4+18+√5+2=4789 E=1103+√20+√5+√2=3812 Total Energy Configuration A: E(rAD=133.67
� � � � � � � � Compute Energy of Config. A • Energy of initial configuration E1 = 1 10 1 + 5 + 18 + 20 = 20.95 E = 1 10 2 + 5 + 5 + 5 = 26.71 2 E = 1 10 4 + 18 + 5 + 2 = 47.89 3 E4 = 1 10 3 + 20 + 5 + 2 = 38.12 Total Energy Configuration A: E({rA})= 133.67
Boltzmann Probability PI Number of configurations A(P-N)!P=#Ofso25N1=637560 NEf of atoms =4 What is the likelihood that a particular configuration will exist in a large ensemble of configurations? Boltzmann probability P( E({r} r))=exp depends on energy and temperature
Boltzmann Probability P! Number of configurations = - P=# of slots=25 NR ( P N )! NR=6,375,600 N=# of atoms =4 What is the likelihood that a particular configuration will exist in a large ensemble of configurations? ({ } ) = exp Ø Œ -Er Boltzmann probability Pr ({ }) ø depends on energy º kT ß œ and temperature B