A simple direct quantum model which, with no random phase assumptions and with arbitrary initial conditions, evolves to the Boltzman distribution
aa r X i v : . [ phy s i c s . g e n - ph ] J un A simple direct quantum model which, with norandom phase assumptions and with arbitraryinitial conditions, evolves to the Boltzmandistribution
Michael J. Caola6 Normanton Rd., Bristol, BS8 2TY, [email protected] 30, 2020
Abstract
We consider M systems (each an electron in a long square cylinder) uniformly arrangedon a ring and with Coulomb interactions. Exact straightforward numerical time-dependentperturbation calculation of a single N-level ( .
7) system, with no (random) phase assumptions,system show a Boltzman distribution. We exploit the physical ring symmetry and developseveral hierarchical physical equation set so of increasing generality and (computation) speed.Given the impressive history of theoretical quantum-mehanical statistical mechanics, our resultsmight seem surprising, but we observe that accurate calculation of correct physical equationsshould mimic Nature.
An important part of Statistical Mechanics is to explain how a physical system can tend to anequilibrium state, usually based on its energy structure. Most analysis, both classical and quantum-mechanical, invokes probability assumptions which in the latter are additional to those of the basicnon-relativistic Schrodinger wave-function[,,]. We treat the quantum case and in particular wish tounderstand how a quantum system can (in time) tend to a state ψ = N X n =1 c n ψ n (1)whose (eigen)states ψ n and (eigen)energies E n are defined by Hamiltonian H : H ψ n = E n ψ n , ω n = E n / ~ , Ψ n ( r , t ) = ψ n ( r ) e − i ω n t . The expansion coefficient c n = c n ( t → ∞ ) gives the probability w n that in equilibrium the system is in state ψ n : w n = | c n | = e − βE n / X n e − βE n (2)This (2) is the Boltzman distribution where β = 1 /kT , T = temperature and k is the Boltzmannconstant. 1 Analysis
Our model is M identical systems uniformly on a ring radius R and normal x . The model is isolated(from the rest of the universe). Each system is an electron in a ”matchstick” box which is a long a square b ≪ a prism, see fig.1. The physical environment each system m = 1 ..M is that of its M − M systems: every system m = 1 ..M is in the ’same heat bath’. This ’same heat bath’ is essentially a 2 D model with finite M ,and it would seem that in 1 D or 3 D only M = ∞ is possible.The energy of an electron in an infinitely deep potential rectangular box, sides ( a x , a y , a z ), is E n x ,n y ,n z = s a x a y a z n x a x + n y a y + n z a z ! (3)and eignfunctions, ω = E/ ~ ,Ψ n x ,n y ,n z ( x, y, z, t ) = ψ n x ,n y ,n z ( x, y, z ) e − iω nx,ny,nz t (4) ψ n x ,n y ,n z ( x, y, z ) = s a x a y a z (cid:18) sin πn x a x x sin πn y a y y sin πn z a z z (cid:19) (5) ≡ ψ n x ( x ) ψ n y ( y ) ψ n z ( z ) (6)Our matchstick system with { a, b ≪ a } means that energies ( E N, , − E , , ) ≪ E , , : the N lowestenergy states enjoy a large energy separation from higher ones and it is a well-established quantumcommon-place that the effects of a perturbation on the system may be accurately calculated usingthese N states only — we have a N -level system. Thus,with n x ≡ n , we henceforth deal only withstates ψ n, , , n = 1 ..Nψ n ( x ) = ψ n, , ( x, y, z ) = r ab (cid:16) sin πna x sin πb y sin πb z (cid:17) (7)The last part of the geometry is, classically stated, the separation r m,m ′ of electrons at positions x m in system m from x m ′ in system m ′ , see Fig.1: r m,m ′ ( x m , x m ′ ) = r ( x m − x m ′ ) + 4 R sin (cid:16) π ( m − m ′ ) M (cid:17) (8) If there are no interactions between systems then system m has wave-function (1)Ψ m = N X n =1 ¯ c mn Ψ n (9)where probability ¯ w mn = | ¯ c mn | is independent of time t , but otherwise arbitrary. In natural fact theelectron e m experiences the Coulomb repulsion V m of its M − e m ′ = m : dV m = N X m ′ = m e m de m ′ r mm ′ (10)In Eq.(10) we treat e m = e classically as a point-charge electron, and e m ′ quantum mechanically: de m ′ = e | ψ m ′ ( x m ′ ) | dx m ′ is the infintesimal charge within dx m ′ of system m ′ , so with r mm ′ ofEq.(8), 2 m = N X m ′ = m e m Z a de m ′ r mm ′ = e N X m ′ = m Z a | ψ m ′ ( x m ′ ) | dx m ′ r ( x m − x m ′ ) + 4 R sin (cid:16) π ( m − m ′ ) M (cid:17) (11)With, c.f. Eq.(1,9), Ψ m ′ = N X n =1 c m ′ n Ψ n (12)we finally have that the time-dependent perturbation on system m = 1 ..M caused by its M − V m ( x m , t ) = e N X n ,n M X m ′ = m c m ′ n ( t ) c m ′ n ( t ) ∗ Z a Ψ m ′ n ( x m ′ , t )Ψ m ′ n ( x m ′ , t ) ∗ dx m ′ r ( x m − x m ′ ) + 4 R sin (cid:16) π ( m − m ′ ) M (cid:17) (13) We summarise nearly 100 year-old time-dependent QM perturbation theory, with general H , ψ n and V ( t ). The exact solution to the TDSE i ~ ∂ Ψ /∂t = ( H + V ( t ))Ψ is Ψ = P n c n Ψ n , where [L&L]˙ c n = 1 i ~ X n ′ c n ′ Z Ψ ∗ n ′ ( r , t ) V ( r , t )Ψ n ( r , t ) d r . (14)With correspondance ( H , ψ n , Ψ n , c n , V, d r ) ⇒ ( H m , ψ mn , Ψ mn, , , c mn , V m , dxdydz ), substitution ofEqns(4,7,8) into Eqn(14) and evaluation of the R R ..dydz integral give˙ c mn ( t ) = 4 e a i ~ N X n ′ ,n ,n e i ( ω n − ω n ′ + ω n − ω n ) t c mn ′ ( t ) M X m ′ = m c m ′ n ( t ) c m ′ n ( t ) ∗ Z a Z a dxdx ′ sin( πnx/a ) sin( πn ′ x/a ) sin( πn x ′ /a ) sin( πn x ′ /a ) r ( x − x ′ ) + 4 R sin (cid:16) π ( m − m ′ ) M (cid:17) (15)Also, to conventionally manipulate the differential set Eq.(15), we need to replace the integer pair( m, n ) by the single integer p : counting in mixed base ( M, N ) gives p = m + ( n − M . Eq.(15) thenbecomes ˙ c p ( m,n ) = 4 e a i ~ N X n ′ ,n ,n c p ( m,n ′ ) M X m ′ = m c p ( m ′ ,n ) c p ( m ′ ,n ) ∗ e i ( ω n − ω n ′ + ω n − ω n ) t Z a Z a dxdx ′ sin( πnx/a ) sin( πn ′ x/a ) sin( πn x ′ /a ) sin( πn x ′ /a ) q ( x − x ′ ) + 4 R sin ( π ( m − m ′ ) /M ) (16) p = p ( m, n ) = m + ( n − M m = 1 ..M n = 1 ..N p = 1 ..M N
To solve/use Eq.(16) we must give them initial values ¯ c p = c p ( t = 0), let them solve until time t,and then examine the probabilities w p ( t ) = | c p ( t ) | . A typical choice of initial values is ¯ c p = 1 / √ N :all initial probabilities are equal, which incidentally corresponds to a temperature T = ∞ .Note that Eq.(15) is non-linear, c p c p ′ c p ′′ ∗ occurs on the r.h.s. We are still using the traditionallinear Schr¨odinger equation Eq.(14) and the non-linearity arises from our formulation § V ( t ). 3f we put d mn ( t ) = c mn ( t ) e − iω n t , use ˙ c mn e − iω n t = ˙ d mn + iω n d mn in (16), and revert d → c , we obtainthe ’Interaction Representation’ (IR):˙ c p ( m,n ) = − iω n c p ( m,n ) + 4 e a i ~ N X n ′ ,n ,n c p ( m,n ′ ) M X m ′ = m c p ( m ′ ,n ) c p ( m ′ ,n ) ∗ Z a Z a dxdx ′ sin( πnx/a ) sin( πn ′ x/a ) sin( πn x ′ /a ) sin( πn x ′ /a ) q ( x − x ′ ) + 4 R sin ( π ( m − m ′ ) /M ) , (17)which may be useful in numerical evaluation. We shall see that solutions w p ( t ) of Eq.(16) are often “steady-state oscillatory”. We assume thatany physical measurent must last a (small) finite time ∆ t , so that any measured or inferred physicalprobability W p ( t ) is an an average over many “oscillations”, W p ( t ) = 1∆ t Z t +∆ tt w p ( t ′ ) dt ′ (18)One might call (16) coarse-graining, a form of RPA: but this occurs at time t , and there is no RPAfrom 0 → t with evolution following (16). M individual systems, c mn This is Eq.(16) which we recall for convenience:˙ c p ( m,n ) = 4 e a i ~ N X n ′ ,n ,n c p ( m,n ′ ) M X m ′ = m c p ( m ′ ,n ) c p ( m ′ ,n ) ∗ e i ( ω n − ω n ′ + ω n − ω n ) t Z a Z a dxdx ′ sin( πnx/a ) sin( πn ′ x/a ) sin( πn x ′ /a ) sin( πn x ′ /a ) q ( x − x ′ ) + 4 R sin ( π ( m − m ′ ) /M ) (19) p = p ( m, n ) = m + ( n − M m = 1 ..M n = 1 ..N p = 1 ..M N. M similar systems, c n We now exploit the physical symmetry of our model to simplify the complex amplitude c mn . We startall systems with the same initial probability, w mn (0) = w n (0) and assume/postulate that this holdsfor all future t , w mn ( t ) = w n ( t ). Since the environment of each site m = 1 ..M is the same and eachinteracts with the M − w mn = w m ′ n = w n .With c mn ≡ r mn e iθ mn (pure math polar form, real r and θ , no phys.), this means that w n = ( r mn ) ≡ r n .We postulate that θ mn = θ n ( t ) + θ m (0) where θ m is time-independent, reflecting the static invarianceof our ring geometry. We now have c mn ( t ) = r n ( t ) e iθ n ( t ) e iθ m whose substitution in Eq.(15 or 16)eliminates e iθ m : c n ( t ) = r n ( t ) e iθ n ( t ) (20)We can show that in Eq.(16) P Mm ′ = m → P M − m ′ =1 (independent of m ) , we ’normalise’ R a → R ,finally giving ˙ c n = 4 e i ~ N X n ′ ,n ,n c n ′ c n c ∗ n e i ( ω n − ω n ′ + ω n − ω n ) t Z Z M − X m ′ =1 dxdx ′ sin( πnx ) sin( πn ′ x ) sin( πn x ′ ) sin( πn x ′ ) q a ( x − x ′ ) + 4 R sin ( πm ′ /M ) (21) m = 1 ..M n = 1 ..N .4.3 P3: Infinite M similar systems, c n We are interested in a system in an infinite M → ∞ heat-bath, and the sum M − X m ′ =1 q a ( x − x ′ ) + 4 R sin ( πm ′ /M )in Eq.(20) can be approximated by an integral R ∞ ..dm ′ which Mathematica 7 gives as an Elliptic EK = EllipticK (cid:18) − R a ( x − x ′ ) (cid:19) . Eq.(20) then becomes ˙ c n = 4 e i ~ N X n ′ ,n ,n c n ′ c n c ∗ n e i ( ω n − ω n ′ + ω n − ω n ) t Z Z dxdx ′ sin( πnx ) sin( πn ′ x ) EK (cid:18) − R a ( x − x ′ ) (cid:19) sin( πn x ′ ) sin( πn x ′ ) (22) n = 1 ..N We have implemented the above analysis in Mathematica 12 to give numerical results, particularlythe measured probability W p ( t ) = 1∆ t Z t +∆ tt w p ( t ′ ) dt ′ . (23)A typical result for initial ¯ c p = c p ( t = 0) = 1 / p ( N ), N = 3 and M = 3 of actual (instaneous) w p ( t ) and measured (experimental) W p ( t ) probabilities is shown in Fig.2. Visually, the intantaneousoscillatory chaos is converted to experimentally observed constant results.Numerically, with energies E n ∼ n the ratio ( E − E ) / ( E − E ) is 0.6 theoretically; physicallywe use (2)and Log ( w n ), see Fig.2, which give (0 . − . / (1 . − .