Typicality paradigm in Quantum Statistical Thermodynamics
Barbara Fresch, Giorgio Moro
Dipartimento Scienze Chimiche
Università di Padova
Outline
1) The framework: microcanonical statistics
versus the dynamics of an isolated quantum system
2) Equilibrium properties and fluctuations
3) What populations? Statistical Ensemble
4) Typicality
5) Thermodynamics
6) Conclusions: an open issue and further developments
1) The framework
Dynamics of a quantum pure state for an isolated system (no interactions
and entanglement with the surrounding, otherwise the system’s
wavefunction does not exist)
H : Hilbert space
Ĥ: Hamiltonian operator for the energy
Hamiltonian (energy) eigenvalues Ek and eigenstates Ek
Hˆ Ek = Ek Ek
for k = 1, 2,
Schroedinger equation for the evolution of the wavefunction Ψ (t )
∂
i Ψ (t ) = Hˆ Ψ (t )
with Ψ (t ) | Ψ (t ) = 1
∂t
Density matrix operator: ρˆ (t ) := Ψ (t ) Ψ (t )
Quantum observable a(t ): expectation value of operator Aˆ
a(t ) := Ψ (t ) Aˆ Ψ (t ) = Tr { Aˆ ρˆ (t )}
Partition of the overall system as subsystem+envinronment:
reduced density matrix µˆ (t ) of subsystem
µˆ (t ) := Trenvir. { ρˆ (t )}
1) The framework
Microcanonical density matrix operator
ρˆ µC
:
Ek ρˆ µC Ek ' = δ k ,k ' ρ kµ,Ck
µC
for Emin < Ek ≤ Emax
µC  = 1/ N
ρ k ,k 
otherwise
= 0
where ∆E := Emax − Emin is small enough but not too much
(Emax ⇒ internal energy U of Thermodynamics)
µC
Conventional interpretation of ρˆ : average of ρ (t ) amongst a
collection of system’s copies
1) The framework
In recent years:
1) quantum computer theory
2) decoherence program (Zurek)
Statistical Thermodynamics of a single system evolving according
to the Schroedinger equation
2) Equilibrium properties and fluctuations
Quantum dynamics in a finite dimensional Hilbert space H N defined by
an upper energy cut-off Emax
H N := span { Ek | Ek ≤ Emax }
Wavefunction expansion on the energy eigenstates
N
Ψ (t ) = ∑ ck (t ) Ek
ck (t ) = Ek | Ψ (t ) = e − iEk t / ck (0)
k =1
− iα k ( t )
(
)
=
e
c
t
P
Polar representation of the coefficients: k
k
2
Populations: Pk = ck (t ) = ck (0)
2
(constants of motion!)
Phases: α k (t ) = α k (0) + Ek t / Pure State Distribution (PSD): homogeneous distribution of the phases for
a given set of populations.
2) Equilibrium properties and fluctuations
TN
∆ N −1
B
2) Equilibrium properties and fluctuations
Randomly perturbed harmonic oscillators: results for the ground state
element of the reduced density matrix
0.5204
0.4756
µ1,1
µ1,1
0.4754
0.5202
0.4752
0.52
0
1
2
3
t
8000
4
x 10
0
1
2
4
3
t
4
x 10
8000
5
4
4000
4000
0.4751
5
0.4753
0.4755
µ1,1
0.5199
0.5201
µ1,1
0.5203
Two different (random) choices of the populations (or of Ψ (0) )
2) Equilibrium properties and fluctuations
Equilibrium properties from the asymptotic time average
a := limτ →∞
1
τ
∫
τ
0
a (t )dt = Tr { Aˆ ρˆ } = a ( P)
N
ρˆ = ∑ Pk Ek Ek
k =1
Equilibrium properties depend on the choice of the populations!
Note: PSD allows the calculations of fluctuations about
the equilibrium value
Population dependence also for the (Shannon) entropy
S := k B ∑ k =1 Pk ln(1/ Pk )
N
and the (internal) energy
N
U := Ψ (t ) Hˆ Ψ (t ) = ∑ k =1 Pk Ek
3) What populations?
No way of imposing populations to a macroscopic system!
How to get predictions about a system if the populations are unknown?
The populations can be characterized only at the statistical level!
Statistical Ensemble for the populations
that is
Sample Space D + probability density p ( P ) for the
populations P = ( P1 , P2 , , PN )
Sample space: simplex in N dimensions
(∑
N
)
P =1
k =1 k
Average of f ( P ) on the Ensemble:
1
1− P1
0
0
f = ∫ dP1 ∫
dP2 ∫
1− P1− PN −2
0
dPN −1 [ f ( P ) p( P ) ]P
No a priori information on p ( P ) !
N =1− P1− PN −1
3) What populations?
From the lack of knowledge
Random Pure State Ensemble (RPSE): homogeneous distribution of Ψ (0)
on the unit sphere in H N
z
=
Ψ (0)
y
x
x, y, z = real or imaginary
parts of ck (0)
3) What populations?
(
From the (Euclidean) measure in H N: p ( P ) = ( N − 1)!δ 1 − ∑ k =1 Pk
N
)
Note: RPSE is defined only in a finite dimensional Hilbert space
necessity of the high energy cut-off Emax
From p ( P ) : statistical sampling of the populations and of the system’s
(equilibrium) properties
3) What populations?
Example for randomly perturbed harmonic oscillators
800
-4.5
n = number of oscillators
700
-5
8
600
( )
p µ1,1
-5.5
σµ
1,1
-6
500
σf =
-6.5
400
-7
300
5
n=5
0
0.31
6
7
100
0.315
0.32
n
7
8
6
0.325
0.33
0.335
µ1,1
f
)
= variance of f ( P)
-7.5
200
(f −
2
0.34
0.345
0.35
0.355
0.36
3) What populations?
Different realizations of the system lead to different values of the
equilibrium properties, in particular of the Entropy and of the Internal Energy
What a relation with Thermodynamical properties?
Note: standard microcanonical analysis corresponds to a specific choice
of the populations
µC

1/
N
for Emin < Ek ≤ Emax
=
µC
Pk 
otherwise
= 0
4) Typicality
Concept of typicality from Information Theory.
A is a typical event in a broad meaning if
i) Prob { A} = 1 − ε with ε <<1, that is Prob { A} =ε
Strong form of typicality if
ii) Prob { A} = 1
like the measure/probability of irrational numbers in the interval [ 0,1]
In our case:
i)
Pro b { µ11 − µ11 ≤ 2σ µ11 } 0.99
ii)
Lim n→∞ Prob { µ11 − µ11 ≤ K } = 1 ∀ K > 0
Qualitatively speaking, the overwhelming majority of systems has a
typical value of the property.
4) Typicality
The same behavior is found for the internal energy and the entropy
Entropy Distribution for systems of n spins
n = 15
-1
10
σ
-2
10
n = 14
-3
10
n = 13
-4
10
0
5
n
10
15
n = 11
n = 10
0.645
n = 12
0.650
0.655
0.660
0.665
Entropy per spin S/n
For large enough systems, the variability of populations does not
influence substantially the observables which, then, can be identified
with their Ensemble averages
S
U
µˆ
5) Thermodynamics
Is the RPSE description consistent with the equilibrium thermodynamics
of macroscopic system?
The answer is positive since, for a generic system characterized through
its density of states
g ( E ) = δ ( E − Ek )
∑
the following items are derived
k
a) Typicality in the thermodynamic limit (number of components n → ∞ )
which allows the identification of the entropy ( S ), of the internal
energy ( U ) and of the properties of a component trough µ̂
b) Entropy equation of state S = S ( U ), by eliminating the dependence
on the energy cut-off Emax
S = f ( Emax )
U = g ( Emax )
S = f ( g −1 ( U ))
5) Thermodynamics
c) Both S and U are extensive properties therefore the Temperature
is an intensive parameter
1 d S
:=
T d U
d) S = S ( U ) is a convex increasing function of U , therefore both
S and U are increasing functions of the Temperature
e) The canonical form for the reduced density matrix of subsystem
µˆ ∝ exp {− Hˆ ss / k BT }
Hˆ ss : Hamiltonian of the subsystem
6) Conclusions
An open issue
1) There are other possible Statistical Ensembles for the populations.
In the past the Fixed Expectation Energy Ensemble (FEEE) having the
constraint < Ψ | Hˆ | Ψ >= E = constant, has been proposed as the quantum
counterpart of the classical microcanonical statistical mechanics.
2) Consistency with Thermodinamics, points a) to e), as the main
criterion of choice between Ensembles.
As a matter of fact, FEEE does not satisfy conditions d) and e)!
3) Further conditions to be satisfied?
Thermalization
Switch on the interaction Hamiltonian
+
1 spin
9 spins
H = H1 + H E + λ H SE
Solve the Schrödinger equation
ψ S ( 0) = β
ψ E ( 0 ) = ∑ k =1 Pk eiα M
kmax
k
ψ (0) = ψ S ( 0 ) ⊗ψ E ( 0 )
ψ E (0)
from RPSE
0
-0.05
5
Emax
-0.1
-0.15
0
E
z
M (0) ∝ H E = 0
Mpzz
-0.2
-0.25
-0.3
-0.35
-5
-0.4
M z (0) ∝ H S =
= Tr ( S Z µ ( 0 ) ) = −0.5
-0.45
-0.5
0
500
1000
t
1500
5
ψ S ( 0 ) = 0.34eiα β + 0.66eiα α
1
2
ψ E (0)
from RPSE
0
E
z
M (0) ∝ H E = −0.66
M zS (0) ∝ H S = 0.16
Thermalization
Emax
-5
One observes relaxation
toward
a well defined equilibrium state
(i.e. within the statistical
uncertainty due to the finiteness
of the system)
The equilibrium state
depends
only on the total energy of the
system,
not on the initial state and
equipartition of the energy is
observed
0.1
0
-0.1
Mpzz
-0.2
-0.3
-0.4
-0.5
0
500
1000
t
1500