Dissipative motion in state spaces

124 Pages·1981·5.59 MB·English

by  Alberti, Peter M., Uhlmann, Armin

Dro Peter M. Alberti
Born 1947 in Dippoldiswalde« Studied Physics at the Karl-Marx-
University, Leipzig, from 1966 to 1971; research student from 1971
to 1973o Received Dr* rer. nat. in 1973, Dr» sc. nat. in 1979. He
belongs to the permanent staff of the Dep. of Physics, Karl-Marx-
University-, Leipzig.
Fields of interest: W*-Algebra Theory and Statistical Physics.
Prof. Dr. Armin Uhlmann
Born 1930 in Chemnitz. Studied Mathematics at the Karl-Marx-Univer-
sity, Leipzig, from 1949 to 1954; research student from 1954 to
1957. Received Dr. rer. nat. in 1957, Div. habil. in 1960. From 1958
to 1960 at the University of Jena, since 1960 at the Karl-Marx-Uni-
versity, Leipzig. 1972 member of the Academy of Sciences of GDR.
1973 National Prize. Several years at the Joint Institute for
Nuclear Research, Dubna.
Fields of interest: Algebras with Involution, General Relativity,
Quantum Field Theory and Statistical Physics.
Manus
shoul
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TEUBNER-TEXTE zur Mathematik - Band 33
Herausgeber / Editors: Prof. Dr. Herbert Kurke, Berlin
Prof. Dr. Joseph Mecke, Jena
Prof. Dr. Hans Triebei, Jena
Dr. Rüdiger Thiele, Halle
Peter M. Alberti - Armin Uhlmann
Dissipative Motion in State Spaces
Dissipative motions in state spaces of commutative C#- and W*-Algebras
are examined. A theory of the transformation of states by stochastic
(linear) maps is developed and discussed. The applications include
the characterisation of certain kinds of evolutionary processes in
the state spaces corresponding to systems of classical statistical
physics.
Es werden dissipative Bewegungen in Zustandsräumen kommutativer
C*- und W*-Algebren untersucht. Dazu wirH systematisch eine Trans­
formationstheorie von Zustände, unter stochastischen (linearen) Ab­
bildungen entwickelt und diskutiert. Die dabei gewonnenen Ergebnisse
finden unter anderem Anwendung bei uci >akterisierung bestimmter
Typen von Zeitentwicklungen im Zustandsraum von Systemen der klas­
sischen Statistischen Physik.
On étudie ici les mouvements dissipatifs dans les espaces d'états
des C*- et des W*-algêbres commutatives. On développe ensuite systé­
matiquement une théorie transformationnelle des états relativement
à des applications stochastiques (linéaires) que l'on soumet à dis­
cussion. Les résultats ainsi obtenus s'appliquent, entre autres, à
la charactérisation de développements temporels de types déterminés
dans les espaces d'états des systèmes des la physique statistique
classique.
Иследуются трансформации пространств состояний коммутативных С*- и
ДО*-алгебр. Результаты используются для описание определённых типов
временной эволюции в пространствах состояний классической статисти-*
ческой физики.
2
P R E F A C E
Let us consider a linear transformation of the n-dimensional real
vector space with a distinguished base into itself, mapping each
probability vector onto a probability vector. Then the transformation
and the matrix of the transformation is said to be stochastic.
One could try to generalize this setting: Given a linear space, L ,
and a convex set, K , contained in that space, ask for the linear
maps T : L — » L which transform K into K .
This, obviously, is much too general for obtaining any remarkable
result. In asking for a good choice we look for convex sets, K ,
which, eventually, may serve as a space of states for a physical sys=
tern, i.e. a set of states a la GIBBS and VON NEUMANN, i.e. mixtures,
mixed states (in the terminology of many textbooks).
Then a stochastic map may be identified with (the result of) a motion
of these states respecting its convex structure, i.e., respecting
the performing of new GIBBSian mixtures out of GIBBSian mixtures.
Stochastic mappings of this kind may be viewed as describing the
change of the states, the physical system admits, in the course of
time. This change will be dissipative generally. It involves some
elements of irreversibility if not, by chance, the motion is an auto=
morphism of K • motion's deviation from an automorphism,
its irreversible or dissipative character, will be "measured** by
"entropy-like" functionals.
We then shall pose an inverse problem. Given a path, a trajectory,
in the state space, is there a semigroup of stochastic mappings
generating it.
But how to decide with respect of what convex sets, K , we have good
reasons for considering them candidates of state spaces of physical
systems? There is, of course, no answer once and forever. But the
"algebraic approach" to Quantum Theory and Statistics selects out
and points toward state spaces of C*- and W^-algebras. (Clearly,
there are further essential and unavoidable examples in the state
spaces of certain op^-algebras of unbounded nature. The insight into
the geometry of these state spaces, however, is not yet sufficient
for our purposes.)
At this place we like to stress that about the first half of our text
does not require knowledge of C -theory.
3
Choosing for K the state space, SA , of a C*-algebra A , every
affine map of the state space is induced by a linear map. Thus we
&
fall back to the study of linear transformations, T : A --^ A ,
mapping the state space into the state space.
Having reached this point we have to distinguish rather sharply
commutative and arbitrary algebras: What appears trivial for the
former unfolds a rich structure in the non-commutative case, and what
is almost untractable at present in the latter situation can be sol=
ved for commutative algebras.
What we are knowing concerning the non-commutative case we have col=
lected in our book "Stochasticity and Partial Order" [9] •
The state of affairs as seen by us concerning state spaces of commu=
tative C -algebras we try to explain on the following pages.
As to "technical remarks” we mention: Within every § we start again
in marking equations by (1), (2), ... . We refer to equation (10)
of § 1.2 by (1.2-10) • "Theorem 1.8.4.” , as an example, refers to
the 4th theorem within § 1.8. References are given either by their
number of appearance in the reference table or by that together with
the author’s name. Sometimes we use the symbol V abbreviating
"for every ... it is ... " . In writing := instead of = we like
to stress that we are defining the left-hand side by the right-hand
side. The symbols ST and St are used simultaneously for sets of
stochastic maps.
For valuable discussions we like to thank the participants of the
seminar on Mathematical Physics of the Karl-Marx-University, Leipzig,
in particular B. Crell, G. LaBner, P.Richter, ^.Schmiidgen, W.Timmermann.
We thank A.V/ehrl, Vienna, for cooperating with us on several aspects
of partial order in state spaces; Th. seligman, Mexico-City, who
discussed with us the partial ordering of pairs of probability vectors
and a related "non-commutative" problem; W. Ebeling, Berlin, who ex=
plained to us connections of our investigations with the theory of
dissipative structures. One of us (P.M.Alberti) is grateful to
I. Prigogine and his coworkers, Bruxelles, for exciting discussions
on current research in non-equilibrium Thermodynamics.
Leipzig, December 1980 P. M. Alberti, A. Uhlmann
4
C o n t e n t s
1. Chapter
How to compare finite-dimensional probability vectors
and some related more general problems.
§ 1.1. Notations and definitions 7
§ 1.2. How to compare finite discrete probability
distributions 11
§1.3« An example: Heat conduction between n bodies of
equal heat capacity 18
§ 1.4. Partial ordering m-tupels of vectors 2'
§1.5* Auxiliary constructions 24
§ 1.6. Heat conduction between bodies of different heat
capacities 28
§ 1.7* Master equations 33
§1.8. An inverse problem for master equations 38
§ 1.9* Comments 41
2. Chapter
--------- *
Stochastic maps in commutative C -algebras
§ 2.1. Definitions and notations 48
§ 2.2. K-functionals 52
§ 2.3* Tiie n-tupel problem. Density theorems 54-
§ 2.4. On extremal points of certain convex sets of
stochastic maps 58
3. Chapter
Stochastic maps and h-convex functionals
§ 3.1. Properties of h-convex functionals 58
§ 3*2. h-convexity and the n-tupel problem 60
§ 3*3* An integral representation for the functionals Sf 82
4. Chapter
Standard examples
§ 4.1. Transformations by finite dimensional stochastic
matrices 66
5
§ 4.2. The extreme points of some convex sets or
stochastic matrices 69
§ 4.3. The n-tupel problem with infinite-dimensional
stochastic matrices 76
a
§ 4.4. Doubly stochastic maps acting on L (0,1) 79
5. Chatter
Stochastic dynamics in 1 - spaces
§ 5-1. Definitions. Posing the problem 85
§ 5.2. An outline of the results 88
§ 5-3. A theorem concerning extensions of dynamical
semigroups 90
§ 5.4. The existence of dynamical semigroups generating
trajectories fulfilling conditions (H) and (R) 9 4
§ 5-5. Further proofs of theorems announced in § 5*2. 97
§ 5.6. Another aspect of the condition (H) 102
§ 5-7. Remarks 105
§ 5.8. Appendix 108
References 113
6
l*_Hpw_tp_compare_finite-dimensional_probabilit2_vectors_and
1,1, Notations and definitions.
In the first chapter we are dealing mainly within finite-dimensional
real linear spaces. Let us denote by
¿n ( or simply by 1 )
the n-dimensional real vector space the elements of which are de=
noted by a, b, or, more explicitely, by
a = a , a , ... , a J , (1)
A
and in which a norm, the 1 -norm, is defined by
1(311* = U1| + U 21 + ... + lan| • (2)
The real numbers a*5 , j = 1,...,n , occuring in (1) are called
components. Sometimes we write (a)** for the component of a .
The vector a is called positive iff all its components are non­
negative, As usual b ^ a is equivalent with the positivity of
b - a • If all the components of a are strictly positive we call
a strictly positive.
The trace of a vector is by definition the sum of its components;
tr, 3 = a +a +,.,+a • (3)
A vector of trace one the components of which are non-negative is
called probability vector for obvious reasons- In particular
-(b) := {1/n» 1/ll> •••» V b J W
is a probability vector which is refered to occasionally as the
equipartition of ,
Definition 1.1.1.: A linear transformation
is called stochastic iff it is both, trace preserving and
positivity preserving.
If in addition a stochstic transformation fulfils
T £(n) = !(m) (6
the transformation is called doubly stochastic.
With other words stochasticity of T means firstly (T a) >0
if only a > 0 ("positivity preserving”),and secondly for all b
tr.(Tb) = tr,b ("trace preserving").
7
We rephrase the expression " T is a transformation” often by saying
” T is a map”, or ” T is a mapping”, i.e. the words ”transformation”,
»»map”, and "mapping” denote the same mathematical structure.
Every linear map (5) can be characterized uniquely by a matrix,
T = (t^) , i = '1,...,n, k = 1,... ,m ,
defined by
^ a * in1 s ( T â )k = ZL a1 • (7)
Stochasticity of T is equivalent with
V i,k s tki > 0 and V i : £. t\ = 1 . (8)
|(
The first condition of (8) gives the positivity of the mapping T ,
and the second guaranties the conservation of the trace.
The additional condition (6) for double stochasticity reads
Ÿ k : 21 *^1 = n / m . (9)
4
In the most important case n = m double stochasticity expresses
itself in the matrix representation by 1) all entries are non-
negativ, 2) the sum of every column and the sum of every row is
equal to 1 .
The infimum of all numbers B fulfilling
V- â « in1 : j|( T a )||„ é B llatt„ (10)
1 *1
is called the norm of T , or the 1 -norm of T . The 1 -norm of T
is denoted by || T and is given by
= sup Z I t\ I • (11)
Occasionally we shall denote the set of all stochastic maps from
in’ infc0 Im b*
Stn,m or* if n = m * by stn s= stn,n *
Clearly, Stn m is a compact set if considered as a subset of the
finite-dimensional space of all transformations (5) •
Later on we need suitable generalizations of the concepts mentioned
above for (commutative) C*-algebras and their state spaces. Here
we include only the straightforward generalization to 1^ •
This space is defined to consist of all sequences
a = (a1,a2... (12)
\
that
U a U := I a* l < O0 • (13)
1 L3
8