D.H.E. Gross and E.V. Votyakov: Phase transitions in “small” systems
125
8 Conclusion
a thermodynamic limit it might well be possible that the
character of the transition changes towards larger system
size.
Micro-canonical thermo-statistics describes how the en-
tropy s(e, n) as defined entirely in mechanical terms by
Boltzmann depends on the conserved “extensive” vari-
ables: energy e, particle number n, angular momentum L
etc. It is well-defined for finite systems without invoking
the thermodynamic limit. Thus in contrast to the con-
ventional theory, we can study phase transitions also in
“small” systems or other non-extensive systems. In this
simulation we could classify phase transitions in a “small”
system by the topological properties of the determinant
of curvatures d(e, n), equation (16) of the micro-canonical
entropy-surface s(e, n):
The great conceptual clarity of micro-canonical
thermo-statistics compared to the grand-canonical one is
clearly demonstrated. Not only that, we showed that the
micro-canonical statistics gives more information about
the thermodynamic behaviour and more insight into the
mechanism of phase transitions than the canonical ensem-
ble: About half of the whole {E, N} space, the intruder of
S(E, N) or the region between the ground state and the
\
line APmB in Figure 4, gets lost in conventional grand-
canonical thermodynamics. Without any doubts this con-
tains the most sophisticated physics of this system. We
emphasized this point already in [28] there, however,
with still limited precision. Due to our refined simula-
tion method this could be demonstrated here with uni-
formly good precision in the whole {E, N} plane. Finally,
we should mention that micro-canonical thermo-statistics
allowed us to compute phase transitions and especially
the surface tension in realistic systems like small metal
clusters [17]. Our finding clearly disproves the pessimistic
judgement by Schro¨dinger [35] who thought that Boltz-
mann’s entropy is only usefull for gases. A recent appli-
cation of micro-canonical thermo-statistics to thermody-
namically unstable, collapsing systems under high angular
momentum is found at [36].
– A single stable phase by d(e, n) > 0.
– A transition of first order with phase separation by
d(e, n) < 0. The depth of the intruder is a measure of
the inter-phase surface tension [34,22]. This region is
bounded by a line with d(e, n) = 0. On this line Pm
is a critical end-point where additionally v1 · ∇d = 0
in the direction of the eigenvector of d(e, n) with the
largest eigenvalue λ1.
– There, the transition is continuous (“second order”)
with vanishing surface tension, and no convex intruder
in s(e, n). Here two neighboring phases become indis-
tinguishable, because there are no interfaces. However,
[
we found a further line (PmC, critical) with v1·∇d = 0
which does not border a region of negative d(e, n). Pre-
sumably d(e, n) should be 0 also. This needs further
tests in other systems. It may also be that these lines
signalize transitions of first order in other, but hidden
conserved degrees of freedom.
D.H.E.G. thanks M.E. Fisher for the suggestion to study the
Potts-3 model and to test how the multicritical point is de-
scribed micro-canonically. We thank H. Jaqaman for critical
reading. We are gratefully to the DFG for financial support.
– Finally a multi-critical point Pm where more than two
phases become indistinguishable by the branching of
several lines with d = 0 or with v1 · ∇d = 0 to give a
flat region with additionally ∇d = 0.
References
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Our classification of phase transitions by the topolog-
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s(e, n) is close to the natural experimental way to iden-
tify phase transitions of first order by the inhomogeneities
of phase separation boundaries. This is possible because
the micro-canonical ensemble does not suppress inhomo-
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emphasized already by Gibbs [21]. Inter-phase boundaries
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the entropy surface. With this extension of the definition
of phase transitions to “small” systems there are remark-
able similarities with the transitions of the bulk. More-
over, this definition agrees with the conventional definition
in the thermodynamic limit (of course, in the thermody-
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We believe, however, that the various kind of tran-
sitions discussed here have their immediate meaning in
“small” and non-extensive systems independently whether
they are the same in the thermodynamic limit (if this then
exist) or not. For systems like the Potts model that have 12. Chapter VII of [8], footnote on page 75.