Złożoność, przypadek, samoorganizacja

Transkrypt

ZŁOŻONOŚĆ, PRZYPADEK,
SAMOORGANIZACJA:
Plus ratio quam vis
Paradygmaty ewolucji naukowo-kulturalnej
w pracach Ilyi Prigogine’a
Ewa Gudowska-Nowak
Instytut Fizyki im. M. Smoluchowskiego i Centrum Badania Układów Złożonych
im. M. Kaca, Uniwersytet Jagielloński w Krakowie
Studium Generale Universitatis Wratislaviensis, 28.02.2012
1
wtorek, 13 marca 2012
„Nasza wizja przyrody ulega gruntownej zmianie
ewoluując ku wielości, czasowości i złożoności...
Zasada wzrostu entropii opisuje świat jako uklad
przechodzący stopniowo do chaosu. Tymczasem ewolucja
systemów biologicznych czy społecznych dowodzi, że to,
co złożone wyłania się z tego, co proste...
Brak równowagi , którego wyrazem jest przepływ
materii i energii, może być źródłem porządku...”
I.Prigogine, I. Stengers „Z chaosu ku porządkowi: Nowy
dialog człowieka z przyrodą”, PIW, 1990
lIya Prigogine 1917-2003
Nagroda Nobla 1977
Tworzenie jest podstawową cechą Wszechświata, a
kreatywność naukowa równoległa do kreatywności
artystycznej ...
(z wywiadu z E.Lévy, Le Point, 18.01,2002)
2
Samoorganizacja i uporządkowanie
3
• Samoorganizacja jest spontanicznym procesem
tworzenia się uporządkowanych struktur
przestrzennych i czasowych
• Układy samoorganizujące się są emergentne
(całość posiada własności odmienne od własności
układów składowych)
4
Układy złożone: narzędzia opisu
Ẋ = σ(Y − X)
Ẏ = −XZ + rX − Y
teoria układów
dynamicznych,
Ż = XY − bZ
projekt Wolfram/Mathematica
procesy stochastyczne,
A. Bulsara, Physics Today, 1993
termodynamika
nierównowagowa
ui = kui −
k � u3i
+ ε(ui+1 + ui−1 − 2ui ) + q sin ωt +
K. Emanuel, Physics Today, 2006
5
tial evaporation rates, the air a short distance above the sea
surface must be much drier than would be the case were it in
equilibrium with the sea.
√
Dξi (t)
The hurricane as a
Carnot heat engine.
This two-dimensional
plot of the thermodynamic cycle shows a
vertical cross section of
the hurricane, whose
storm center lies along
the left edge. Colors
depict the entropy distribution; cooler colors
indicate lower entropy.
The process mainly
responsible for driving
the storm is the evaporation of seawater,
which transfers energy
from sea to air. As a
result of that transfer,
air spirals inward from
A to B and acquires
entropy at a constant
temperature. It then
undergoes an adiabatic expansion from B to
C as it ascends within
the storm’s eyewall. Far from the storm center, symbolically
between C and D, it exports IR radiation to space and so
loses the entropy it acquired from the sea. The depicted com-
Układy złożone: narzędzia opisu
teoria macierzy przypadkowych
teoria grafów/sieci
Table
Examples of basic RNA
structure
motifs. Fromoftop
to bottom:
a single
strand
Bon,1: Vernizzi,Orland,
Zee,secondary
Topological
classification
RNA
structures,
Journal
(PDB 283D [5]), a helical of
duplex
(PDB 405d
[6]), a379
hairpin
stem and
loop (PDB 1e4p [7]), a bulge
Molecular
Biology
(2008),
p 900
(PDB 1r7w [8]), a multiloop (PDB 1kh6 [9]). From left to right: spacefill view, three-dimensional
structure, secondary structure motif. The pictures are made with MolPov [10], Jmol [11] and PovRay
[12].
3
Hufnagel, Brockmann, Geisel, Rorcast and control of epidemics, PNAS, 101
(2004) p. 15
6
Fig. 1. Global aviation network. A geographical representation of the civil
aviation traffic among the 500 largest international airports in #100 different
Arises
in statistical mechanical
models
e.g.
BIFURKACJE
w teorii
układów
dynamicznych
of magnets
or nieliniowych
neural networks
W układach
zmiana parametru kontrolnego skutkuje pojawieniem się
nowych rozwiązań stacjonarnych
Plot y = x and y = β tanh x
Example 3.4.1 ẋ = −x + β tanh x
3.4.1 ẋ = −x + β tanh x
statistical mechanical models
Fig. 3.4.3
ets or neural networks
W pobliżu punktów bifurkacji istotną rolę
zaczynają odgrywać fluktuacje: mogą one
Arises
in ostatistical
mechanical
models e.g.
decydować
wyborze gałęzi
stabilnych stanów
of
magnetsdostępnych
or neuralwnetworks
stacjonarnych
dynamicznej ewolucji
układu...
e.g.
Plot y = x and y = β tanh x
x and y = β tanh x
Fig. 3.4.3
Poszukiwanie rozwiązań stacjonarnych: w prostych
przypadkach wystarczą metody graficzne,
współcześnie pełne rozwiązania problemów
wielowymiarowych otrzymujemy numerycznie...
Fig. 3.4.4
Fig. 3.4.3
13
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% t X"A! ! B#1 " X#X
Y #&
X, E 64 05
n the Brusselator.
We
discuss and obtain a
AND
C.
PÉREZ-GARCÍA
PHYSICAL
REVIEW
B.
PEÑA
ed analytically with direct numerical simulations of the
itude equation,
including
spatial modulasselator
model. Sec.
VI summarizes
our conclusions.
2
2
thogonality
are use
nforms appearing in the model. This equa% t Yconditions
"BX!X"Fredholm
Y #D&alternative#
Y,
each order. In the neighborhood of the bifurcation point
tained in many chemical models for stripe
critical amplitudes A j follow the so-called normal fo
II.
THE
REACTION-DIFFUSION
MODEL
the hexagonal case has not been discussed
TheirYgeneral
form
canconcentrations
be derived from standard
techniq
where X and
denote
the
of activato
of symmetry-breaking
bifurcations
%16'. A normal
form
#8–10$.
From symmetry
arguments,
substrate, respectively.
Here D
is a parameter
proportio
The
Brusselator
is considered
one some
of the FIG.
simplest
1. Turing
patterns
direct
simulations
of Brussin
scribes
perfect from
extended
patterns,
but slight variations
ed
that
it
must
include
some
nonpotential
/D
and,
aswith
us
theinstadiffusion
theincluded
two Phys.
species
tion-diffusion models exhibiting Turing and Hopf
B.ratio
Pena,can
C.ofPerez-Garcia,
Rev.
E. D
64of
Y(2001)
X 056213
pattern
be
byinitial
means
spatial
terms
,
!0.04
hexagons,
"b#
,
!
model
(A!4.5,D!8):
"a#
besides
the
usual
diffusive
linear
one
ies. The spatiotemporal evolution of the main variables
is keptisas the
control
parameter
of theat problem.
suitable
symmetries,
so one arrives
the so-called ampli
striped
pattern,
and
"c#
,
!0.98
reentrant
hexagons.
FIG. 1. Turingtechnique
patterns fromwe
direct
simulations of Brusselator
s of
multiple-scale
compute
equations. Westeady
discuss state
in the of
following
the form of is
t
en
byathe
following
partial
differential
equations: The homogeneous
these
equations
model (A!4.5,D!8): "a# , !0.04 initial hexagons, "b# , !0.30
equations for different planforms.
f the amplitude
equation for the Brusselator.
"(X
#9$. Let
briefly
recall he
ply
u
striped pattern, and "c# , !0.98 reentrant hexagons.
s
s ,Y
s )"(A,B/A)
Stripes
are characterized
by us
a single
amplitude
A
2
2
2 hexagons
2
ons of this% X"A!
equation! B#1
are "stripes,
B"1#!
A around us . Consid
X#X Y #& X,
results of the
linear
stability
evolves
according
to analysis
t
2
2
L!
t phases, mixed modes,B"1#!
and distorted
Ahexa2
2 . arrives at
small
perturbations
u"(x,y)
in
Eqs.
!1"
one
!1"
2
"B
"A #D! 1
L!the stability of2 these
"3#
2 .
tability analysis sets
2
2 #D!
"A
( 0 . t A! , A"g # A # 2 A# / 20 . x #
. 2y A,
% t Y "BX!X"B
Y #D&
Y,
2ik c
ns in regard to amplitude disturbances.
1 mode
B 2
Perturbations
u
are
expanded
in
normal
Perturbations
u
are
expanded
in
normal
modes
u
2
ity limit is obtained from phase perturbawhich
is the normal
form
for a supercritical
bifurcation
x
%
u"Lu#
#2Axy#x
y
,
t
$
u
exp
&
(k)t#ik•r
leading
to
the
eigenvalue
equa!
%
'
re
X
and
Y
denote
the
concentrations
of
activator
and
Pierwowzór
reakcji
oscylacyjnej
zaproponowany
przez
# k# !k
0
&(k)t#ik•r
to variations,
the eigenvalue
A ' leading
!1known ase
cmuch
# k# !k c u0 exp
a %term
that accounts
for spatial
mplitude
varies
more rapidly
in time! $
2
" (Here
& (k)#)!0
in
which ( and
) indicate the
tion &i(k)
strate,
respectively.
D
is
a
parameter
proportional
to
Prigogine’a
Lefevera
2
Newell–Whitehead–Segel
equation
%17' in)convection
and, therefore,
it
becomes
enslaved
by
the
&
(k)
"
(
&
(k)#)!0
in
which
(
and
indicate
tion
trace
and
the
determinant
of
the
linear
operator
L,
respecnomena.
usual, BL is the linearized operator
diffusion
ratio
of48
the(1968)
two species
D Y /D X and, as where
J. Chem.
Phys.
1695slight
hases.
The
slow
dynamics
spatial
tively.
Depending
on theofvalues
of the
parameters
Aand
and D
trace
the determinant
of thebylinear
operator sets
L, ofres
Squares are formed
two perpendicular
ept as thethecontrol
parameter
of
the
problem.
system will undergo a Turing "stationary# bifurcation or a
with equal
wavevalues
numbers,of
i.e.,the
"a# kparameters
tively.
Depending
on
the
an
1!k2 and "b# # kA
1# !
The G.
homogeneous
steady
state
of
these
equations
is
simHopf
instability.
As
a
matter
of
fact,
2D
reaction-diffusion
Nicolis, I. Prigogine „Selforganization in Nonequilibrium
symmetry inPhysical
Fourier sp
They display four ©2001
peaks with
D4American
4!5"/056213!9"/$20.00
64
056213-1
The
S
the
system
will
undergo
a
Turing
"stationary#
bifurcation
systems
that
display
Turing
patterns
also
present
a
Hopf
bi,Y
)"(A,B/A)
#9$.
Let
us
briefly
recall
here
the
us "(X
Systems”
When condition "b# does not hold one obtains rectangula
s
s Wiley, New York, 1977
!1/D
The latteranalysis
gives risearound
to oscillations
if * +
Hopf
instability.
As patterns;
a matter
of fact,
bimodal
instead,
without2D
the reaction-diffu
condition "a# rho
.
Considering
lts of thefurcation.
linear stability
u
s
2
$( !1#A "1)/A %14'. Otherwise, a Turing instability
takes
"rhombicTuring
cells# arise.
Squares
or present
rhombs should
o
systems
that
display
patterns
also
a
Hop
ll perturbations
u"(x,y)
in
Eqs.
!1"
one
arrives
at
place, the case we focus on hereafter.
coupled equations of the kind: ( 0 . t A 1 ! , A 1 "g # A 1 #
furcation.
latter
gives rise to oscillations if * + !
to a curve
BThe"g
Theukład
marginal
condition
Nieliniowy
reakcji
i dyfuzji( & --!0)
dla leads
krytycznych
wartości
0 # A 2 # A 1 , in which g 0 depends on k1 •k2 "valid for an
2
!B(k) kontrolnych
the minimum
of which yields
the$(
critical
point
/3).
not around
1fluktuacje
!
B 2 2 obserwowane
1#A
"1)/A
%14'.2 1Otherwise,
a Turing instability
parametrów
są
gęstości
2
*#2Axy#x
) , !A * ' . To describe
naturally
the sepa(B
Hexagons are built up by three modes satisfying the r
x
%
u"Lu#
y
,
!2"
c ,k c )! % (1#A
t
zmieniające
się
periodycznie
w
czasie
i
przestrzeni
place,
the
case
we
focus on
hereafter.
A critical values the rescaled
!1 parameter , nant condition
ration from the
k1 #k2 #k3 !0 "resonant triad#, with
Brusselator
!
!
"
"! "
!
!
!
"
"
"! "
!0)break
leads
to a symm
curv
marginal
( &modes
!(B"B c )/B c dubbed supercriticality, replaces BThe
as the con! # k 2 #condition
! # k 3 # . When the
the rotational
9
trol parameter.
# ! # k 3 # ) squeezed
hexagons
but not reflexions
( # k 1 # 2 # k 2yields
!B(k)
the
minimum
of
which
the
critical
rewtorek,
L 13ismarca
theBefore
linearized
operator
2012
going to a perturbative analysis let us show exsult. Distortions breaking rotation and reflexion symmet
Gdzie zawodzi tradycyjna
termodynamika?
•Aksjomatyczna, „tradycyjna” termodynamika jest
[WHERE THERMODYNAMICS FAILS]
CAUTION:
BOTH HOT AND COLD
elegancką, CONTENTS
samozgodnąMAY
teoriąBEmatematyczną.
•Kluczowe pojęcia: stan, funkcja stanu, równanie stanu
Temperature seems like such a simple, universal concept. Things may be hot or cold, but they always have a temperature, right?
Not quite. It is possible to assign a temperature only to systems (such as the molecules in a glass of water) that are in, or almost in, a stable
condition known as equilibrium. As systems deviate from equilibrium, the temperature becomes progressively more ambiguous.
(wiąże ze sobą parametry
stanu). SEVERE DISEQUILIBRIUM
MODEST DISEQUILIBRIUM
•Infnitezymalne, adiabatyczne zmiany stanu są
odwracalne w czasie
EQUILIBRIUM
A glass of water, left undisturbed, comes to room
temperature. The water molecules collide with one
another and reapportion their energy so that their
overall pattern of velocities stabilizes. Although the
glass contains billions on billions of molecules, it
takes only one number— the temperature — to describe this pattern. Classical thermodynamics applies.
Heating the water from below disturbs the equilibrium. But if the heating is modest, individual layers of
water remain approximately in equilibrium — socalled local equilibrium — and the water can be described by a temperature value that increases from
top to bottom. The theory of nonequilibrium thermodynamics developed in the 20th century applies.
If you crank up the heat, individual layers may no longer
be even approximately in equilibrium. The molecules become a chaotic jumble in which the concept of temperature ceases to apply. To describe the system, you would
have to introduce a raft of new variables and, in the
most extreme case, specify the molecular velocities one
by one. This situation demands a new theory.
[WHY A NONEQUILIBRIUM THEORY IS NEEDED]
ORDER FROM DISORDER
Although the molecules
in a system out of equilibrium may be hopelessly jumbled, the system can become ordered in other ways.
MOLECULAR
VELOCITIES
Fraction of
DYSSYPACJA NIE MUSI PROWADZIĆ DO
CHAOSU!
THE SECOND LAW
A common example is when you put ice in a of the four laws of thermodynames, students apply thermodynamics to dynamic
glass of water. The ice melts, and the water in the ics, the study of heat and energy.
systems such as car engines to calculate quantiglass reaches a uniformly lower temperature. If Whereas the first law states that
ties such as efficiency. But these applications
you cannot get something for nothyou zoom in to the molecular level, you find an ing, the second law states that you
make an implicit assumption: that we can apintense activity of molecules frantically moving cannot even get something for
proximate a dynamic process as an idealized
about and endlessly bumping into one another. something. Almost all processes
succession of equilibrium states. That is, we
In equilibrium, the molecular activity organizes lose some energy as heat, so to get
imagine that the system is always in equilibrium,
itself so that, statistically, the system is at rest; if something, you have to give someeven if the equilibrium shifts from moment to
thing more. Such processes are
some molecules speed up, others slow down, irreversible; to undo them exacts a
moment. Consequently, the efficiency we calcumaintaining the overall distribution of veloci- toll in energy. Consequently:
late is only an upper limit. The value that engines
ties. Temperature describes this distribution; in
reach in practice is somewhat lower because they
fact,
the
very
concept
of
temperature
is
meaningN Engines are inherently limited in
under nonequilibrium conditions.
particles is brought down to an average (if slight- chines is very deep. Fluctuationsoperate
of the chemical
their energy efficiency.
fully
only
when
the
system
is
in
equilibrium
or
sufThe
second
law describes how a succession
fluctuating) value. Although a few isolated energy affect a molecular motor in the same
ficiently
near
it.
of
equilibrium
states
can be irreversible, so that
events may show completely unpredictable be- way that a random and variable amount of fuel
N Heat pumps tend to be more
Thermodynamics
therefore
deals
only
with
the
system
cannot
return
to its original
havior, a multitude of events shows a certain reg- efficient
affectsthan
thefurnaces,
piston because
of a carthey
motor. Therefore, the
MORE
TO state
situations
of
stillness.
Time
plays
no
role
in
it.
In
without
exacting
a
price
from
its
surroundings.
ularity. Therefore, quantities such as density can move
longrather
tradition
of applying
than generate
heat. thermodynamics to
EXPLORE
reality,
of course,
nature
never stands
still,For
andthis large motors can be extended toAsmall
melted
ice Alcube does not spontaneously refluctuate
but remain
predictable
overall.
ones.
Non-equilibrium
Thermodynamtime
does the
matter.
Everything
is intoa rule
constant
form; you need
it in the freezer,
at a cost
computer
memory
is an
reason,
second
law continues
over the N Erasing
though
physicists
have
other mathematical
toolsto put
ics. S. R. de Groot and P. Mazur.
state
of
flux.
The
fact
that
classical
thermodyirreversible
act,
so
it
produces
heat.
in
energy.
To
quantify
this
irreversibility,
the
world of the small.
for analyzing such systems, those tools can be
J.M. Rubi, Scientific American, 2008
10
64From
S C I E NSteam
T I F I C A MEngines
ERICAN
to Molecular Motors
tricky to apply. The equations of fluid flow, for
example,
require researchers
specify the con© 2008
SCIENTIFIC
AMERIC AN,toINC.
ditions at the boundary of a system precisely— a
±
Dover, 1984.
Thermodynamics
N o v e“beyond”
mber 2008
Local Equilibrium. José M. G. Vilar
DANIELA NAOMI MOLNAR
W stanach odległych od równowagi, traci
sens „zwykła” definicja/interpretacja
temperatury
Tradycyjna termodynamika nie opisuje
przejść pomiędzy metastabilnymi stanami
pośrednimi
Teoria nie opisuje uporządkowania w
stanach nierównowagowych
Classical
thermodynamics,
cannot account for that, but the newly developed nonequilibrium theory can.
Some
textbooks
define temperature based as it is on equilibrium,
molecules
as the average random velocity of molecules. In fact, temperature is the measure of
an entire pattern of velocities. MODEST
EQUILIBRIUM
INCREASING
SEVERE
EXTREME
In modest departures from equilibrium,
An unheated glass of
DEPARTURE
DEPARTURE
DEPARTURE
DEPARTURE
this pattern is merely shifted, but in
water at room temperaA glass of water heated
If the temperature gradiAs the heating increases, the
As the heating increases
severe departures, it is distorted,
ture looks the same in
Molecular
velocity
units)
develops
a (arbitraryent
is larger, the water
pattern of convection cells
still further, the chaos berendering temperature meaningless. from below
every direction, a symtemperature gradient. If
begins to overturn, seteventually breaks down into
comes equally distributed
metry known as
the gradient is too slight
ting up an orderly pattern
turbulent chaos.
and the fluid recovers the
isotropy.
to overcome viscous resisof convection cells.
lost isotropy.
the two are in thermal equilibrium.
that
namics is limited to equilibrium situations may
tance toFrom
motion,
the fluid
remains static.
point on, nothing changes.
come as a surprise. In introductory physics classThe second law is the best known
Warunki równowagi i teoria liniowej
odpowiedzi układu
Sν (Eν1 , Eν2 , ...)
U1
U2
Stotal =
�
Sν (Eν1 , Eν2 ...)
ν
U1 ⊕ U2
izolowane
Ej = E1j + E2j = const
przy warunkach δS1 (E1j ) + δS2 (E2j ) = 0
⇒
�
∂S1
∂S2
−
∂E1j
∂E2j
�
δE1j = I1j − I2j ≡ 0
11
δE1j + δE2j = 0
∀δE1j
POZA STANEM RÓWNOWAGI
„siły termodynamiczne”
Xj ≡
�
∂S1
∂S2
−
∂E1j
∂E2j
�
δE1j = I1j − I2j �= 0
siły termodynamiczne generują przepływy
� ∂S dE1j
dS
=
dt
∂A
dt
1,j
j
W konsekwencji, produkcja entropii
wyraża się prostym iloczynem sił i
strumieni termodynamicznych
Rozwinięta przez Onsagera (~1932) teoria liniowej
odpowiedzi (słuszna dla słabych sił X) przewidywała
Φj =
�
k
IIga zasada narzuca
warunek
j
�
dS
Lkj Xk ⇒
≈
Lkj Xk Xj
dt
k
dS
≥0
dt
12
dE1,j
= Φj
dt
�
=
X j Φj
Termodynamiczna analiza stabilności: Kryterium
stabilności Glansdorffa - Prigogine’a nierównowagowych
stanów stacjonarnych w oparciu o „nadmiarową
produkcję entropii”
2
∂ 2
(δ S)N SS
∂t
(δ S)N SS < 0
�
=
δJk δXk > 0
k
I. Prigogine, Science 201, 777 (1978) Nobel Lecture
kryterium lokalne, wystarczający (ale nie konieczny)
warunek lokalnej stabilności stanu stacjonarnego
13
To illustrate we use a characterization of the Ising model [77].
Imagine a 2D grid of lights of varying brightness (from off to
maximum), where brightness is a function of two variables. One is a
random noise factor (individual to each light) and the other is a
neighbor conformity factor whereby each light tends towards the
brightness of its four nearest neighbors on the grid. These two
variables are weighted together to determine the brightness of each
light. In this illustration, the correlation function measures the degree
to which lights have equal brightness levels as a function of their
distance apart on the grid. If noise is heavily weighted, then
interactions are somewhere between independent and interdependent. Weak interactions can result in short-range correlations (Figure I
green) that decay exponentially with distance. Stronger interactions
can result in long-range correlations that decay more slowly (Figure I
pink) (i.e. as an inverse power of distance). The correlation function
can also be defined for distances in time, with an analogous
comparison between weak (short-range) versus strong (long-range)
interactions. No interactions can result in uncorrelated noise (Figure I
grey), and integrating over uncorrelated noise results in a random
walk (Figure I brown).
Prawa skalowania:
układy złożone - przebieg zjawisk obserwowany jako szereg
czasowy
szereg czasowy
funkcja korelacji
widmo mocy
Figure I. Four example time series are plotted in the left-hand panel: random samples from a normal distribution with zero mean and unit variance (i.e. white
noise, in grey), a running sum of white noise (i.e. brown noise, also known as a random walk, in brown), 1/f noise (i.e. pink noise, in pink) and an autoregressive
Kello average
et al,Trends
in Cognitive
moving
(ARMA,
in green), where each sampled value is a weighted sum of a noise sample, plus the previous noise value, plus the previous sampled
Sciences,
14,
p.223-232
(2010)
value. Idealized autocorrelation
functions are shown in the middle panel for each of the time series, where k is distance in time. Note that white noise (i.e. pure
independence) has no correlations, ARMA has short-range correlations that decay exponentially with k, 1/f noise has long-range correlations that decay as an
Rozkłady potęgowe
inverse power of k and brown noise has correlations that decrease linearly with k. Idealized spectral density functions (where f is frequency and S( f) is spectral
obserwowane
dla!1wielu
power) are shown in the right-hand panel in log–log coordinates. White, pink and brown noises correspond to straight lines
with slopes są
of 0,
and !2, whereas
różnych
zjawisk
(rozkład
ARMA plateaus in the lower frequencies.
długości czasów trwania trzęsień
−α
ziemi, energii wybuchu, masy
lawin, etc...)
range correlations in collective measures of component
activities. These predictions are supported by the evidence
reviewed here for neural avalanches [62], power-law distri-
S(f ) = const × f
Szum
różowy:
envelope can be illustrated by tracing a line from peak to
14
peak along a given waveform (i.e. the convex hull). Interwtorek,
13 marca 2012
estingly,
the temporal scaling of amplitude fluctuations in
Układy samoorganizujące się są dyssypatywne
- charakteryzuje je pobór i straty energii
Uporządkowanie obejmuje całość układu, a
dynamiczna zmienność jest zdeterminowana
fluktuacjami parametrów kontrolnych
Układy samoorganizujące się cechuje wysoki
stopień adaptacji do warunków zewnętrznych
i wysoka odporność na zniszczenia
15
Niespójność? Determinizm mikroskopowych praw natury
i kierunkowość zjawisk ewolucyjnych w przyrodzie
Czas jest iluzją...
A. Einstein
[...]Im głębiej analizujemy naturę czasu, tym lepiej
rozumiemy, że trwanie oznacza inwencję, tworzenie
form, stałe doskonalenie absolutnie „nowego”...
Czas rzeczywisty (fizyczny) i czas
subiektywny (durée)
Henri Bergson 1859-1941
Nagroda Nobla 1927
[...]W naturze istnieje wielkość, która zmienia się w tym samym sensie
we wszystkich procesach naturalnych...
Byłoby absurdem twierdzić, że zasada wzrostu entropii jest wynikiem
naukowej obserwacji, niedoskonałości eksperymentalnego pomiaru...
Max Planck 1858-1947
Nagroda Nobla 1918
16
using solely the microscopic dynamical laws: the above
time symmetry implies that if (a, b, c, d) is a possible
ordering so is (d, c, b, a).
The explanation of this apparent paradox, due to
Thomson, Maxwell and Boltzmann, as described in
references [1−13], shows that not only is there no conflict
between reversible microscopic laws and irreversible
macroscopic behavior, but, as clearly pointed out by
Boltzmann in his later writings1), there are extremely
strong reasons to expect the latter from the former.
These reasons involve several interrelated ingredients
which together provide the required distinction between microscopic and macroscopic variables and
explain the emergence of definite time asymmetric
behavior in the evolution of the latter despite the total
absence of such asymmetry in the dynamics of the
former. They are:



■ the great disparity between microscopic and
macroscopic scales,
■ the fact that the events we observe in our world are
determined not only by the microscopic dynamics,
but also by the initial conditions of our system, which,
if taken back far enough, inevitably lead to the initial
conditions of our universe, and
■ the fact that it is not every microscopic state of a macroscopic system that will evolve in accordance with
the entropy increase predicted by the second law, but
only the „majority“ of such states − a majority which
however becomes so overwhelming when the number
Dlaczego
nie
obserwujemy
ewolucji
verned
by
Hamiltonian
dynamics
with
some
specified
blem and
w czasie?
Hamiltonian H(X) whichodwróconej
we will assume
for simplicity
Jakie są
przyczyny
symetrii
to be an even function
of the
momenta:łamania
no magnetic
stinctly different
time t ,
fields. Given H(X), the microstate
X(t),
odwrócenia
watczasie?
sn’t unspill, eggs
determines the microstate X(t) at all future and past
s break but do not
times t during
which
the system
be or was isolated:
różnice
skal mikrowill
i makro
•istotne
układów
dynamicznych
determinowana
ver younger. These
X(t) = Tt–t X(t•ewolucja
). Let X(t
) and
X(t + τ), with
τ posi- jest nie tylko
przez prawa mikroskopowe, ale także przez warunki
in time − they
tive, be two such
microstates. Reversing (physically or
początkowe
efine the arrow of
we obtain
mathematically)
all velocities
at time
t + τ,będzie
mikroskopowy
stan układu
ewoluował
•nie każdy
zgodnieIfz we
zasadą
entropii,
a jedynie większość
that the relevant
a new microstate.
nowwzrostu
follow
the evolution
for spośród
takichτstanów
no such distinction
another interval
we find that the new microstate at
laws permit all pro- time t + 2τ is just RX(t), the microstate X(t) with all
42 Journal 6, (2007)
J. Lebowitz,
Physik
. This
leads
to a great velocities reversed:
41-46 (M.Planck Medaille)
it it why don‘t we
RX = (r,–p, r, – p, ..., rN, – pN).
esses run backwards?
ring run backwards
Hence if there is an evolution, i. e. a trajectory X(t),
how can time-rever- in which some property of the system, specified by a
les, the microscopic function
in a certain
 f(X(t)),
 behaves

N
Nway as t increaive rise to observed
ses, then if f (X) = f (RX) there is also a trajectory in
veryday world?
which the property evolves in the time reversed direc17
ory, the „theoryi
tion.
i
a
sible motions of atoms and molecules, the m
components of material systems, give rise t
time-irreversible behavior of our everyday
In the context of Newtonian theory, the „
of everything“ at the time of Thomson, Max
Boltzmann, the problem can be formally pr
follows: the complete microscopic (or micro
classical system of N particles is represented
X in its phase space Γ,
b

c
d
Fig. 1 How would you order this sequence of „snapshots“ in time? Each
represents a macroscopic state of a system containing, for example a fluid with
two „differently colored“ atoms or a solid
in which the shading indicates the local
temperature.
Physik Journal 6 (2007) Nr. 8/9
© 2007 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
X = (r , p , r , p , ..., r , p ),
r and p being the position and momentum
Termodynamiczna strzałka czasu
wyznaczana procesami dyfuzji,
wzrastającymi korelacjami i
pojawieniem się zachowań
kolektywnych w układach
wielociałowych
The persistence of memory, S. Dali (1931)
W podejściu (filozofii) Szkoły
Brukselskiej, kierunkowość czasu
jest podstawowym zjawiskiem
wynikającym z dynamiki złożonych
układów fizycznych
Rozszerzenie teorii Poincaré:
układy równań dynamicznych są
niecałkowalne jeśli zawierają
rezonanse pomiędzy rozmaitymi
stopniami swobody
Rezonans: przejściowy stan
metastabilny układu stwarzający
możliwość efektywnego przekazu
energii
18
"2:19-)(2&$,*;"(9(="*>?@*"*>?AB*)".1'(*7+8C*(*%)"j%(2&$,*6-3#2$D(<2"#*%*"2:19$Dj
.(9(*)j3.-;&E*F;(*3#*;"(9(*)&9(%1Dj*"23#.(9$D*%*9-=6<#9'#=*;"(#9*;"-.j$&$,*1:%"(
)*(:,#%D"*9-=4.#9*:-*=($"#.%&*6-%(9-=4.9-)#DC*(*%*9-<#"*3#*%)"j%(2#*%*(93&2(="*"*$&G
NEWS & VIEWS Q&A
3-'%9"#<#3#=*6#2"j*H129$D*)*=-;"<2-È$"*"*="5.($D"*9-=4.#9E*I-%.#51<-)(2"#*3&$,
H129$D"*6.%#%*;"(9(*>?@*"*>?AB*=-ß#*6.-)(:%"o*:-*%(;1.%#¦*=-;"<2-È$"*9-=4.#9
-:$%#6"#2"(*'"*"$,*-:*6-%(9-=4.9-)#D*=($"#.%&C*(*2('362"#*:-*%(6-$%j39-)(2"(*9(2G
Box 1 | The discovery of chromosome territories
$#.-5#2#%&*"*6.-5.#'D"*2-)-3)-.1E
Nurty współczesnej teorii układów złożonych (podejście
synergetyczne) w naukach biologicznych (biologia systemów)
a
Random organization model
Chromosome territory model
Laser
damage
Laser
damage
Prediction
Prediction
b
I
I&$E*JE*K.(H"$%2(*"<1'3.($D(*"23#.(9$D"*=":%&*;"(9(="*7+8*"*;"(9(="*$%-)"#9(E*(L*?"#o*"23#.G
(9$D"*;"(#9*$%-)"#9(*M7G7N*9-<-.*2"#;"#'9"LC*61293&*)*9-<-.%#*$%#.)-2&=*-%2($%(Dj*$%j'3#$%9"
;"(9(*6-*%(9(ß#2"1*7+8E*;L*?"#o*"23#.(9$D"*8G7E*O%&*)*9-<-.%#*$%(.2&=*P*;"(9(*)".1'-)#N
)*9-<-.%#*$%#.)-2&=*P*;"(9(*<1:%9"#N*j$%2"9"*$%#.)-2#*P*"23#.(9$D(*=":%&*;"(9(="*<1:%9"="
"*)".1'-)&="N*)%&*2"#;"#'9"#*P*"23#.(9$D#*=":%&*;"(9(="*<1:%9"="E*>(D)"9'%&*9-=6-2#23
'"#$"*MÈ.-:#9*.&$"2&L*-;#D=1D#*QJR*$%j'3#$%#9*;"(9-)&$,E*O5*:#*+,(''#&*SEC*>(T.(3"<*8EC*U(HH-G
.#(1*VE*MW00/LX*7#6(3"3"'*+*T".1'*"2H#$3"-2*6.-3#"2*2#3)-.9E*Y!-<#$1<(.*?&'3#='*S"-<-5&ZC*[X*W@0
II
0
III
1
2
3(=X)–5
6–8
9–11
F$%#91D#*'"C*ß#*6-%2(2"#*(.$,"3#931.&*"*-54<2&$,*)(È$")-È$"*'"#$"*9-=4.9-)&$,
M. Chorąży, Nauka 1/2011
At the turn of the twentieth century,
occupied distinct territories (a, left panel),
T.
Misteli,
Nature,
445
379-381
(2007)
"*"23#.(9$D"*;"(9-G;"(9-*6-%)-<"*2(*;1:-)(2"#*1-54<2"-2&$,*,"6-3#%*-*=-<#91<(.2&$,
Carl Rabl and Theodor Boveri proposed
localized
damage
would affect only a
that each chromosome maintains its
small subset of chromosomes, whereas if
6-:'3()($,*ß&$"(C*.-:%(D($,*.#51<($D"*#9'6.#'D"*5#24)C*"23#.(9$D"*=":%&*6-:'"#$"(="C
individuality during the cell cycle, and
the chromatin fibres of each chromosome
'3(;"<2-È$"*"*#<('3&$%2-È$"*'"#$"E*+#<#=*;"-<-5""*'&'3#=4)*D#'3*3(9ß#*2-)(*9<('&H"9($D(
Boveri explained this behaviour in terms
were randomly distributed throughout the
19of ‘chromosome territories’.
$,-.4;C*:"(52-'3&9(*"*3#.(6"(E*!-ß<")#C*ß#*1:(*'"*)'9(%(o*3(9"#*)%&*"*6-j$%#2"(*)*'"#G
nucleus (a, right panel), many chromosomes
The existence of chromosome territories
was demonstrated experimentally during
would be damaged. b, Three sets (I–III)
of hamster chromosomes after laser
Podsumowanie i „luźne” wątki...
Powszechność układów złożonych i zjawisk samoorganizacji czasowoprzestrzennej w różnych skalach obserwacji przyrody (od biologii po
kosmologię..) sugeruje, że zrozumienie charakteru „praw” , mechanizmów
odpowiedzialnych za „samoporządkowanie” i spontaniczne powstawanie
struktur może być kluczem do spójnego opisu materii.
Czy (nieliniowa) dynamika, przyczynowość i przypadek (fluktuacje) są
wystarczającymi elementami tego opisu?
20
[...] Przesłanie Jacquesa Monoda mówi: życie, a w szczególności człowiek jest niereligijnym cudem.
Wedlug mnie jest inaczej . Materia posiada potencjał związany z samoorganizacją, z którego właściwie nie
zdawano sobie sprawy w czasach Monoda. Z drugiej strony, przesłanie Sartre’a i Camusa mówi, że
wszechświat jest absurdalny.To poczucie absurdu jest naturalne we wszechświecie, w którym człowiek jest
oderwany od otaczającej go przyrody.Walczyłem z tymi poglądami wykazując, że istnieje pewna
racjonalność, ale że zawiera ona niepewność i strzałkę czasu...
I. Prigogine
(wywiad z E.Lévy, Le Point, 18.01,2002)
21
Dziękuję za uwagę
Plus ratio quam vis
22

Złożoność, przypadek, samoorganizacja

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