Download Systems with negative specific heat

Z. Physik 235, 339-- 352 (1970)
Systems with Negative Specific Heat
W . THIRRING*
CERN, Geneva, Switzerland
Received November 20, 1969
Some systems for which the binding energy increases more rapidly than linearly
with the number of partieles, are shown to exhibit negative specific heat c for some
energies. In thermal contact with larger systems, c < 0 creates an instability, and in the
canonical ensemble one sees only a phase transition. It is argued that supernovae are,
in essence, a phase transition of this origin.
1. A Surprising Theorem that is Simple to Prove, but which is Wrong
It is a fact known to astrophysicists that if radiation energy is extracted f r o m a star whose nuclear fuel is exhausted, the star will contract
and heat up. Thus a star acts like a system with negative specific heat**.
There are proofs that under reasonably general assumptions the specific
heat of interacting particles is positive. In this paper we shall try to
clarify this perplexing situation.
The astrophysical arguments all depend on the virial theorem.
Since the latter comes only from the 1/r behaviour of the potential and
not from its attractiveness, one actually arrives at the more general
Theorem 1.
Theorem 1. According to classical statistical mechanics, every piece
of condensed matter has negative specific heat.
Before proving Theorem 1, we have to specify what we mean by
condensed matter. Condensed means that the system keeps together, for
E < 0 , without being put in a box, and thus we do not have to include
the potential of the box in the Hamiltonian.
By matter we mean a system of N electrons and nuclei with only
static Coulomb interactions. Thus the Hamiltonian of our system is
HN=K+V,
(1)
the kinetic energy K being the usual
N
2
* On leave from the University of Vienna, Vienna, Austria.
** The only reference to this fact that I could find is Landau, L. D., Lifchitz, E.:
Statistical physics, 2nd ed., p. 62. Oxford: Pergamon Press, and D. terHaar,
Proc. Int. Conf. on Statistical Mechanics, Kyoto 1968.
23*
340
W. Thirring:
and the potential
V= ~ ei ej
rtj
(3)
Now the proof is trivial. If ( ) denotes the thermal expectation value,
the virial theorems and equipartition tell us (k = 1)
(_K)=I
(V)=(H)=_E=
- - 3N2
T.
(4)
Thus, the specific heat
dE
c- dT-
3N
2 <0,
(5)
Q.E.D.
Let us first study this absurd result for the simplest case, namely
one particle in a Coulomb potential:
p2
H = 2m
e2
r
(6)
Although one particle is not much of a thermodynamic system, for
which the microcanonical ensemble could be used the above reasoning
should also apply to it. The volume of the phase space under the energy
surface (0 = step function)
f2(E) =5 d3 x d 3p O(E-H)
(7)
is finite for E < 0 and then easily calculated to be ,,~( - E ) --~.
This behaviour comes about since in x-space the volume available
goes ,,~(E)- 3, and from momentum space we get a factor (E) ~. Actually
this also follows immediately in quantum theory where we have E =
-Ry/n 2, and the number of states ~ below a quantum number n goes
as n3"~lEl-~. Thus for the entropy S(E) we have
S(E)=-
ln(-E),
1 dS
T-dE-
3
2E'
(8)
in agreement with Eq. (5) for N = 1.
However, for a real thermodynamic system we need many particles,
and then one soon realizes where the proof of Theorem 1 goes wrong.
First of all we shall need a box, otherwise 0 will be infinite since we
can always send one particle to infinity and lower the energy of the rest.
Even in box
ON= 5 da~rSdaNpO(E-HN)(N!) -1
box
(9)
Systems with Negative Specific Heat
341
diverges if we have several particles. Carrying out the p integration we
have
e . e . \ 3N/2
ON,~fd3Nr E-- Z -*-J|
Fij /
,
(10)
where the integral goes over the region where E > ~ e i e f l r i j . To have
~ # 0 for E < 0 we need an attractive pair (i,j), but then the integral
diverges for small r,j since the integrand then goes as r;) 3me. Thus we
have also to modify the potential at small distances to get a meaningful
expression for ON for which the formal arguments of the proof apply.
Therefore we have to take into account an external virial due to the box,
and an internal virial due to a small distance repulsion. In the next
section we shall try to estimate the effect of the former for the gravitational case (eiej ~ - xmimj) where we have only attractions, and therefore the best chance for a small effect of the box. In Section 3 the question
of the internal virial will be settled for a very truncated form of the
potential for which Or can be calculated exactly.
To conclude this section we shall comment on the quantum theoretic
problem. There ~ N ( E ) = N u m b e r of states below E exists even without
cut-off at small distances. The non-relativistic Hamiltonian (l) has for
arbitrary N a lower eigenvalue Eo, in contradistinction to the relativistic
case where we get a collapse for e~ei~-tcm~rn j and N sufficiently
large. A recent analysis of Dyson 1 and collaborators has shown that
IE o I ~ N is a rather exceptional situation for particular combinations
of statistics and signs of e. In many cases ]Eol increases faster with N.
We shall see later that this leads to interesting thermal consequences.
Regarding Theorem 1, one might think that it applies rather to the quantum theoretic case, since there the problem of the internal virial disappears. However, the equipartition Theorem holds only in the classical
limit, the virial theorem being generally true. Thus there seems to be no
system for which the proof of Theorem 1 holds. Nevertheless we shall
see that Theorem 1 reflects an essential feature of statistical mechanics
for gravitationally interacting systems.
2. A Non-Rigorous Treatment of the Gravitational Case
In this section we shall estimate ~N for ei eJ ~--~: by a method
which has some intuitive appeal and may be appropriate for (unshielded)
long-range forces. The idea is that the main part of the gravitational
potential that a particle feels comes mainly from the bulk of the particles
at large distances rather than from its immediate neighbours. Thus we
shall divide the volume V of the box in M ~ N cells of equal size, large
1 Dyson, F., Lenard, A. : J. Math. Phys. 8, 423, 1538 (1967).
342
W. Thirring:
enough to contain many particles but small enough so that the potential
can be treated as constant inside a cell. Instead of integrating over all
particle coordinates daNr, we shall sum over the number of particles
n, in the cell around the point x= (~ = 1 ... M). This is exact if the integrand is constant for all particle configurations leading to the same
occupation numbers. It is a standard combinational problem to transform d aNr(N!)- i into
.
1__,,,1,"
i
(n2)t
-
1
,,~=o(nMt)
N, Ln=
In this way we obtain (all ms =89 h = 1)
~(E)=(N !)- I S d3S p daS ,'O (E- ~=lp]- ~>jv(r~, rj)) =( N !)- ~S d3N
N
N
"exp{3-~-~
In[E-~>f(ri,r,)]2rce/3N}"O(E--'i~>
f(r,,r,))
oo
n l m O n2 "~ 0
9 exp
oo
/12Mr= 0
ln[E-~n~v(x~,x,)nr
L
-
(11)
~=1
==/i'
M
~ n= In n = M / e V} .
at=l
Here we have used Stirling's formula to the accuracy N! =(N/e) N. So
far our manipulations rest on the assumption that the potential is constant in each cell and n~>>1, which can be achieved by making N and
M sufficiently large. Next we make an approximation that is popular
in statistical mechanics, namely represent ~ by a single term for which
tie*
the summand reaches a maximum as function of the n= subject to the
M
condition ~
n==N:
~t=1
B~
va
8S
~3n~
3N
2
M
(i2)
0
--2 ~ v(x=, x # ) no# + l n -n,- =M
~t/T=const
3N T# =1
V
"
Systems with Negative SpecificHeat
343
Here we have identified temperature as
1
T-
dS
3N
dE - 2
1
M
(13)
E - ~ n~v(x~, x~) np
ot>~
Here the E-dependence of the n ~ does not matter since
~
~So On~
~=1 ~
__~,=~On~
aE = -
#
O N= 0
1 ~E = -u
e---g
(14)
9
The maximum condition (15) is, of course, the barometric formula
n=,,~e -v~
V(x) being the potential due to all particles. This procedure can be justified rigorously in the limit N-+oo if the n,o correspond
to an absolute maximum of S. We shall not attempt to find conditions
on v, which assures a maximum, since in our case it is not true anyway.
However, one might note that not only in the trivial case v =0 but also
for v ( x , y ) = ( r e / N ) ( x - y ) 2 for which the many-body problem can be
solved (12) leads to the exact result. The ease of interest for us is v(x, y) =
- r e / I x - y ] . We know already that then the expression for O diverges
and we shall keep a cut-off at small distances in mind for the case of
trouble. Furthermore we shall pass to the continuum limit
M
n=--~-=p(x,),
~,
M
ax
,=,---,T-fd
M
N = ~, n = = I d a x p ( x ) ,
S= -idaxpo(x)ln--
a=l
po(X)
(15)
(zc T) ~ e ~
for which Eqs. (12) assume the familiar form of the equations of static
equilibrium of a star consisting of an isothermal ideal gas
Po (x) = e (~/T)s d3x' rpo (x')/Ix- x'll const.
(16)
Guided by intuition we shall look for spherically symmetric solutions
and hence use for V the unit sphere. Their Eq. (16) reads (r=[x[)
d r' r' z Po (r') ( 1 / r - 1) + j"d r' r' Po (r') (1 - r')
Po (r) = Po (1) exp
r
(17)
The differential version of Eq. (17)
d po =
T dr
4 ~zre
~. ,
- f i - p o ( r ) j d r r'2po(r ')
0
(18)
344
W. Thirring:
allows us to express the quantities of interest by the asymptotic form
of p:
T p~(1)
N=
V(1) = - N
tc po(1) '
1
4rt T p o ( 1 ) - 3N T = 4 r c T S dr r 3 dp~
o
dr
=-~cSdr'r'24rCpo(r ')
drr24~po(r)=V
(19)
~
0
po(X)
S = - S d 3 x po (x) In (~ T) ~ e§
3N
7
N t2c
2 lnrcT-~N-Nlnpo(1)+~+8~Po(1
).
We are now prepared to approach the central issue, namely whether
the effect of the external virial will upset our conclusion that the system
as a whole has negative e. From Eq. (19) we note
3N
3N
E =-- 88 T + V = - - T + 4 r c Tp0(1 ) .
2
(20)
Here the last term represents the contribution of the external virial, and
the question is whether its temperature derivate will overcome the
- 3 N/2 we had obtained at the beginning. We shall see that this is not
the case; on the contrary, it also contributes negatively. This comes
from the property of a star to heat up when it shrinks, and hence Po (1)
decreases with increasing temperature. To see this one has to use the
well-known solution of Eq. (18) corresponding to a p strongly concentrated at the origin. One finds that p.,~ 1/r2 is a solution for T O=Nx/2,
Eo = - N 2 x / 4 , and for small deviations from these values the solution is
P~
N
{ l + T - T ~ [l+2(r(-l+iVv)/2+r(-1-'VV/2))]}
(21)
With Eq. (22) we then obtain [for I( T - To)/To 1~ 1]
E=
N z rc
4
r3
7 N ( T - To)
nNtr
N
S=N I_/~ln~-ln-~-~--~
1
7 T-T.]
2
2
(22)
T~]--" "
Thus we have actually
clr= To =
- - ~7- N ,
(23)
Systems with Negative Specific Heat
345
F =3/2 In ~7-2~ - tn(3-'~)- 3~.~
0.4
0.2
0
t.L -0.2
-0.4
-0.6
-0.8
II+ H I I P I I I T I I I M I I I I I I I H I I I ] I ]
-2.0
-
1.0
0
-g
1.0
FIM 11 M I I I I I
2.0
Fig. I. F as function of 7 according to formula (24)
being even more negative than according to Theorem 1. Now dearly
this result cannot be exact because we have not used the cut-off without
which there is no S. The fact is that S[po] is not a maximum. Looking
at the second derivative p2S/tpo(x)fpo(X ) one finds by expansion in
spherical harmonics that for l + 0 it is actually a negative definite kernel
but not for l = 0 . This can be seen directly by inserting p(r)=(N/47r).
( 3 - 7)r -r, V<3, into S for E = -NZtc/4. One calculates
S(7)=N fi 3~ l n ~7cN~
_ l n _ 4 _ ~ N+ _ ~ l3n.
7-27
5-27
ln(3_?)_
? ~"
(24)
S,r is actually zero for ? =2, as it has to be since p,,~r-2 is a solution
of Eq. (18). But S,?rl?= 2 =N/3. Thus we do not have an absolute maximum but a relative minimum. What happens is that S is unbounded and
goes to oo for 7 =2.5 (Fig. 1). This shows that entropy favours a strong
concentration at the origin, and we shall encounter this collapse on
further occasions. Our findings are parallel to the classical results of
the instability of an isothermal ideal star. The origin is clearly the need
for a cut-off to get a bounded S. There seem to be ways of modifying
the 1/r potential in order to get stability without introducing too much
internal virial to make c positive. We shall not discuss this here, since in
the next section we shall construct a system with negative c in a much
simpler fashion such that the evaluation of S involves no problems.
346
W. Thirdng:
3. A n E x a c t Solution for a Somewhat Artificial Version of a Star
In this section we shall analyse a system that incorporates the essential features of the previous one, which lead to c < 0 . We shall be
guided by the two conditions 2 which guarantee that for the microcanonical S(E) one has
a2S
{~2 ~2S~-~
0E 2 < 0
and therefore
c--
~
-~-2-]
>0.
They are roughly that at large distances the forces are not repulsive and
that E,,~N. The former condition is satisfied for the gravitational case,
and thus the failure of the second must be responsible for c<0. This
leads us to the following model for a star. Inside an interaction volume
Vo, each particle has an attractive interaction with the other particles
inside the volume and outside Vo the particles are free. With the stepfunction
1 if x e Vo
(25)
Ov~
0 if x~Vo
a non-local potential having this feature is (v > 0 is constant)
v (Xk, Xj) = -- 2 VOvo(Xk) Ovo(Xj).
(26)
In this way the total potential energy is -vN2o, where No is the
number of particles in Vo. The evaluation of ~ is now a simple combinational problem, and we find (if the volume outside Vo is eF. Vo)
(E) = ~
I d aNp daN x 0 [ E - Z p2 _ Z V(x,, x j)]
i
i>j
~3N]2
-
N!(3N/2)! r
V N ~3N/2
-- (3N/2)!
S d aNX (E + V ~ Ov,(X~) Ovi(Xj)) 3N[2
N
~
(E+No2 v)3N/2er(N-No)
No = N.,in
(27)
~'i
No ! ( N - No) !
N
--
~
eS(E'N'N~
So = N.,,~
Thus the expression for ~ is again a sum but in this case over a single
variable. Now it is easy to see when S as a function of No has a maximum.
Indeed (with Stifling)
OS (E, N, No)
3N No v ,
ONo
- E-+~o v - m
No
N-No
F =-0
(28)
has for E<O only one solution since
~2S
dNo~ -
3NvE-3NN2v
(E+No2v) 2
N
No(N_No) <0,
2 Linden, J. van der: Physica 32, 642 (1966).
0 < N o < N , E < 0 . (29)
Systems with Negative SpecificHeat
347
To discuss Eq. (28) we shall introduce "intensive" quantities ct, ~, 0
where, however, the scaling law is different than usual:
E=N2v(z-1),
T=-~NvO,
No=N(1-~).
(30)
From Eqs. (27) and (28) we get a parameter representation for 0 as
function of 5:
0=e-2~+~2>0,
(31)
e=2~_~zq -
3(1-~)
F + l n ( 1 --~)/. "
(32)
If we take a large F ("atmosphere much larger than star") there
will be a sizeable region with Iln (1 - a ) / c t l ~ F. Then we get
0 = 3 ( 1 -e/2),
0 ' ~=
-
3
2F
<0
(33)
and thus again a negative specific heat. On the other hand, near the
minimum of the energy, e ~ 0, cr 0, we have
O=e-2e -3#,
0~1
(34)
and a normal behaviour. This is expected since in this limit all particles
are in Vo and the interaction is just a constant. For E > 0, ~> 1 nothing
guarantees that Eq. (32) determines ~ uniquely, and actually it is not
true for larger F. This is shown by the graph of O(E), which has an
over-hang for e> 1. Of course, in this case one has to choose the
that gives the larger S, and thus bridge the overhang by a vertical line
(Fig. 2). This is verified explicitly by calculating (27) for N=200 on a
computer (Fig. 3). It is interesting to note that the result for N = 1 0
is already close to and for N = 2 5 practically identical with the asymptotic curve. This means that at this point we have a peculiar phase
transition where for constant E the temperature and N O changes
suddenly. Fig. 2 can be described as follows. For large E most particles
are outside Vo and the system behaves normally. On extracting energy,
the system first cools down until suddenly a finite fraction of the particles
fall into Vo. At this moment T jumps up and keeps increasing with
decreasing E. Finally, when most particles are in V0, c again becomes
positive.
One will have noticed that Eq. (28) is just the usual equilibrium
condition of an ideal gas and a similar system with our additional
energy - v N z. One can generalize this for systems with energy -vNL
348
W. Thirring:
0.90
0.60
O
0.30
0o00
0.00
0.30
0.60
,~=2e
0.90
1.20
1.50
Fig. 2. Temperature versus energy according to Eqs. (31) and (32)
oooi - : OoO
/
o
0.00
0.30
0.80
0.90
1.70
~'=2,F=4.5
Fig. 3. Computer evaluation of Eq. (27)
If we minimize the sum of the entropies of such a system and a normal
ideal gas
S = S1 (El, N1) + S2 (E2, N2)
= N i [~ In (E 1 + N~ v) - { In N 1 + In V1]
(35)
+ N2 [-}ln E 2 - ~ ln N2 + ln V2],
subject to N i + N 2 = N , EI+E2=E, we obtain for the corresponding
"intensive" quantities
E = v N r ( e - 1),
N2=c~g,
0=
3T
2Nr-Jv ,
Vz
F = In - ~1-
(36)
the relations:
~(1-~) ~-I
=2__in i-~.+ 2__F_F
~- 1+(1-a)~,
0=e--l+(1-a)
3
e
3
(37)
r.
For 7 = 2 this reduces to Eqs. (31) and (32). For 7 = 1 we have the problem of particles that can fall into an external potential well, a system
which has positive c. However, for larger 7's we get at certain energies
Systems with Negative Specific Heat
349
0.64
0.48
o 0.32
0.16
0
0.00
I
l
I
I
0.30
0.60
0.90
1.20
s
I
1.50
Fig. 4. Temperature versus energy according to Eq. (27)
c < 0 (Fig. 4). This shows that systems whose energy increases more
than linearly with N in particle exchange with normal systems will lead
to negative c.
4. How These Strange Systems Behave in Thermal Contact
So far we have studied the systems with c < 0 using the microcanonical
ensemble assuming that the usual ergodicity arguments that go along
with it also apply. Like for ordinary systems one might hope that adding
a few "grains of dust" may render them sufficiently ergodic in case
they fail to be so from the beginning. Since 8 2 SloPE2< 0, the transition
to the canonical ensemble requires detailed study. Let us first put our System I with c < 0 in thermal contact (exchange of E, not of N) with another
system, System 2. The usual expression for the variation of the total
entropy S with
_
1
_
~?S~
r,
1
=-Ti~~
' c,
~z
ae, !
s (E) = S, (e,) + S2 (E - E,)
5S=SEa
1
( T1
1
T2 )
(3S)
(5Ea)
2 .( C ~. T 2. + C _~~ 2~).
2
tells us the following. Since our 7"/ are positive, we gain entropy for
T, ~=/'2 by transferring energy from the hotter to the colder system. For
Ta =/'2 we have a stable situation if ( 1 / c , ) + ( 1 / c 2 ) > 0 . Thus if both c's
are < 0 we never get a stable equilibrium, and for c2 > 0 only if [ c a [ > c z .
This can be understood as follows. If c a <0, c z < 0 and one system is
slightly hotter it will transfer energy to the other. In doing so it will heat
W. Thirring:
350
S
f
I S
11
I ~
~
/I
TS :~-~-~ E
E
dE
Fig. 5. The phase transition for the region of negative specificheat
up even more, and more energy will be transferred. In this way one gets
further and further away from TI=T2 and equilibrium can only be
reached if one of the c's again becomes >0. This also explains the
instability we found in Section 2 for the pure gravitational case. For
c2 > - cl > 0 and, say, 7"1> T2, energy goes from 1 to 2. Now both
temperatures increase, but 7"1 faster t h a n / ' 2 since Ic~l< c2. Thus again
no equilibrium is reached. Only for cz <1cl I, 7"2 will change faster than
T1, and then an equilibrium is established. For the canonical ensemble,
System 2 would be the heat bath and therefore c2 > Icl 1. Thus there will
never be an equilibrium as long as c~ < 0 and the systems will exchange
energy until Ex is such that cl >0. Hence, in Figs. 2 and 4 the part
with e < 0 will be bridged by horizontal lines. This means that given T
by System 2, the system will jump from the lowest energy where 7"1 = T
to the highest energy where 7"1 =T. The jump occurs when the free
energy on the lower side equals the free energy F on the upper side.
This is evident by plotting S(E) (Fig. 5) where the part with the wrong
convexity is bridged by a straight line since then the upper branch gives
the lower F. This explains why in the canonical ensemble there are no
negative c' s;
F= -Tln
(i
dES(E)e -~/r
)
(39)
always has the right convexity such that
02F
c = - 7' ~
> 0.
_
(40)
What happens in the canonical ensemble is that the region of c < 0 is jumped
over by a phase transition of the first kind. This can be seen for the
Systems with Negative Specific Heat
351
system of w where the canonical partition function can also be calculated along the same lines.
Thus, to define physically the temperature for this system one cannot
use a heat-bath, but one has to use a small thermometer. Then, according
to (38), a maximum of S can be reached for cl <0, ( 0 < c 2 ~ 1 c l I) and
the energy distribution of system 2 will indicate (OY21/OE~)- ~ as temperature.
5. What has this got to Do with Reality
We have noted in the beginning that stars as a whole act like systems
with c < 0. Of course, our considerations cannot be directly applied to
them since they are not isothermal, and have inhomogenous chemical
composition, internal energy sources, etc. Furthermore, quantum effects
will become important. This complicated situation can only be handled
by a computer. Nevertheless our considerations may supply a simplified
model for the dramatic events occurring in the history of a star. For
instance, at a stage where no more nuclear fuel is available which would
burn at this temperature, the core contracts and becomes hotter, giving
its energy to the outer part which expands and becomes colder. This
corresponds just to the heat exchange between two systems with c < 0,
described in the last section. The hotter system is now the core and
the cooler the outer part of the star. This process seems to occur several
times in the lifetime of a star in the formation of red giants or supernovae. These events, although far from being equilibrium phenomena,
reflect the instability of systems with negative specific heat*. Another
system that should reflect these features is a galaxy. Here the stars are
the particles and the dense centre may represent the collapsing phase.
However, our considerations shed no light on the time scale a, 4 governing
these phase transitions. For supernovae where the energy is carried
quickly by neutrinos they are fast, but for galaxies where mainly the
Newton potential transfers the energy they will be very slow.
Appendix
For free particles one sees immediately from Eq. (15) that p ( x ) =
const and therefore iV/11. This gives the classical
* In addition to this thermodynamic instability there is a dynamic instability for
7<4/3(p,-'p~'). For ?=>4/3 the system collapses if heat is extracted, for y<4/3 it
collapses anyway.
3 Prigogine, I., Severne, G.: Physica 32, 1379 (1966).
4 Chandrasekhar, S. : Principles of stellar dynamics. New York: Dover Press 1960.
352
W. Thirring: Systems with Negative Specific Heat
For
N
/s
V=-~~ (x~- xj)2
we first note that this is for N ~ c ~ equivalent to
N
v= Z g.
i=1
The latter is the former plus an harmonic force on the centre of mass,
but one degree of freedom out of N does not matter. Thus we anticipate
the entropy of a 3 N dimensional harmonic oscillator
S=- ~
(zcE)21----NlnN+4N.
x
In ~
(A.2)
The barometric formula (14) is now solved by
tc
po(x)=e-~X2/r. N (---T-~x) ,
which gives for the entropy
3
S(E,N)=jd3xpo(X)[-~-ln~
5
rc x2 -lnN +~3 ln--x---j
Trc ] =
T+-i+-f-
(A.2)
The author would like to thank Dr. I. Wacek for her patient carrying out of
the computer calculations for this work, and to Dr. A. Martin and Dr. S. Epstein
for their help in studying t~2S/rp(x)6p(x'). Furthermore, many colleagues at CERN
helped with stimulating discussions.
Note added in proof. Professor Sciama has kindly pointed out to me that similar
considerations have been published previously by D. Lynden-Bell and Roger Wood,
Monthly Not. Astr. Soc. 138, p,495--525 (1968). Indeed, except for the exactly
soluble model, chapter 3 of this paper, the content of this paper is essentially identical
with the one by Lynden-Bell and Wood.
Prof. Dr. W. Thirring
Theoretical Group
CERN
CH-1211 Genf 23, Schweiz