Rules for Biologically Inspired Adaptive Network Design

Abstract

Transport networks are ubiquitous in both social and biological systems. Robust network performance involves a complex trade-off
involving cost, transport efficiency, and fault tolerance. Biological networks have been honed by many cycles of evolutionary
selection pressure and are likely to yield reasonable solutions to such combinatorial optimization problems. Furthermore,
they develop without centralized control and may represent a readily scalable solution for growing networks in general. We
show that the slime mold Physarum polycephalum forms networks with comparable efficiency, fault tolerance, and cost to those of real-world infrastructure networks—in this
case, the Tokyo rail system. The core mechanisms needed for adaptive network formation can be captured in a biologically inspired
mathematical model that may be useful to guide network construction in other domains.

Full text

DOI: 10.1126/science.1177894
, 439 (2010); 327Science et al.Atsushi Tero,
Design
Rules for Biologically Inspired Adaptive Network
This copy is for your personal, non-commercial use only.
. clicking herecolleagues, clients, or customers by , you can order high-quality copies for yourIf you wish to distribute this article to others
. herefollowing the guidelines can be obtained byPermission to republish or repurpose articles or portions of articles
(this information is current as of January 26, 2010 ):
The following resources related to this article are available online at www.sciencemag.org
http://www.sciencemag.org/cgi/content/full/327/5964/439
version of this article at: including high-resolution figures, can be found in the onlineUpdated information and services,
http://www.sciencemag.org/cgi/content/full/327/5964/439/DC1
can be found at: Supporting Online Material
found at: can berelated to this articleA list of selected additional articles on the Science Web sites
http://www.sciencemag.org/cgi/content/full/327/5964/439#related-content
http://www.sciencemag.org/cgi/content/full/327/5964/439#otherarticles
, 2 of which can be accessed for free: cites 20 articlesThis article
http://www.sciencemag.org/cgi/content/full/327/5964/439#otherarticles
1 articles hosted by HighWire Press; see: cited byThis article has been
http://www.sciencemag.org/cgi/collection/comp_math
Computers, Mathematics : subject collectionsThis article appears in the following
registered trademark of AAAS. is aScience2010 by the American Association for the Advancement of Science; all rights reserved. The title CopyrightAmerican Association for the Advancement of Science, 1200 New York Avenue NW, Washington, DC 20005.
(print ISSN 0036-8075; online ISSN 1095-9203) is published weekly, except the last week in December, by theScience
on January 26, 2010 www.sciencemag.orgDownloaded from

16. J. A. Burns, P. L. Lamy, S. Soter, Icarus 40, 1 (1979).
17. J. A. Burns, D. P. Hamilton, F. Mignard, S. Soter, in
Physics, Chemistry, and Dynamics of Interplanetary
Dust, ASP Conference Series 104, B. A
˚. S. Gustafson,
M. S. Hanner, Eds. (Astronomical Society of the Pacific,
San Francisco, 1996), pp. 179–182.
18. B. J. Buratti, M. D. Hicks, K. A. Tryka, M. S. Sittig,
R. L. Newburn, Icarus 155, 375 (2002).
19. F. Tosi et al., preprint available at http://arxiv.org/abs/
0902.3591 (2009).
20. Besides Iapetus, Hyperion, Titan, and outer-satellite
impacts were suggested; see also (12).
21. Reference (18) mentions an increase of the dust flux by ~20%,
whereas (35) finds as much as a factor of 3 for some cases.
22. D. J. Tholen, B. Zellner, Icarus 53, 341 (1983).
23. The leading sides of the moons beyond Mimas and inside
Titan should be substantially coated by E-ring particles
(24,36,37), making them less useful for this argument.
24. B. J. Buratti, J. A. Mosher, T. V. Johnson, Icarus 87, 3 39 (1990).
25. J. A. Burns et al., Science 284, 1146 (1999).
26. S. S. Sheppard, www.dtm.ciw.edu/users/sheppard/
satellites/satsatdata.html (2009).
27. D. Nesvorný, J. L. A. Alvarellos, L. Dones, H. F. Levison,
Astron. J. 126, 398 (2003).
28. D. Turrini, F. Marzari, H. Beust, Mon. Not. R. Astron. Soc.
391, 1029 (2008).
29. A. J. Verbiscer, M. F. Skrutskie, D. P. Hamilton, Nature
461, 1098 (2009).
30. This idea was developed in several papers
(18,38,39), but under the assumption that dust
from the outer saturnian moons formed Iapetus’
albedo dichotomy.
31. T. V. Johnson et al., J. Geophys. Res. Solid Earth 88, 5789
(1983).
32. T. Denk, R. Jaumann, G. Neukum, in Lisbon
Euroconference Jupiter After Galileo and Cassini,
Abstracts Book 17 to 21 June 2002, Lisbon, Portugal,
abstr. no. P-4.1.18, 2002, p. 118.
33. B. J. Buratti, J. A. Mosher, Icarus 90, 1 (1991).
34. M. E. Davies, F. Y. Katayama, Icarus 59, 199 (1984).
35. K. J. Zahnle, P. Schenk, H. Levison, L. Dones, Icarus 163,
263 (2003).
36. K. D. Pang, C. C. Voge, J. W. Rhoads, J. M. Ajello,
J. Geophys. Res. Solid Earth 89, 9459 (1984).
37. D. P. Hamilton, J. A. Burns, Science 264, 550 (1994).
38. P. C. Thomas, J. Veverka, Icarus 64, 414 (1985).
39. K. S. Jarvis, F. Vilas, S. M. Larson, M. J. Gaffey, Icarus
146, 125 (2000).
40. G. Neukum, B. A. Ivanov, in Hazards Due to Comets and
Asteroids, T. Gehrels, Ed. (Univ. of Arizona Press, Tucson,
AZ, 1994), pp. 359–416.
41. T. Roatsch et al., Planet. Space Sci. 57, 83 (2009).
42. We acknowledge the individuals at CICLOPS (at the Space
Science Institute in Boulder, CO) and JPL (Pasadena, CA), as
well as the members and associates of the Imaging Team for
the successful conduct of the ISS experiment onboard the
Cassini spacecraft. This paper is dedicated to Steve Ostro,
whose work helped considerably to explain the nature of
Iapetus’dark terrain. This work has been funded by the
German Aerospace Center (DLR) and NASA/JPL.
Supporting Online Material
www.sciencemag.org/cgi/content/full/science.1177088/DC1
SOM Text
Figs. S1 to S8
Tables S1 and S2
References and Notes
1 June 2009; accepted 1 December 2009
Published online 10 December 2009;
10.1126/science.1177088
Include this information when citing this paper.
Rules for Biologically Inspired
Adaptive Network Design
Atsushi Tero,
1,2
Seiji Takagi,
1
Tetsu Saigusa,
3
Kentaro Ito,
1
Dan P. Bebber,
4
Mark D. Fricker,
4
Kenji Yumiki,
5
Ryo Kobayashi,
5,6
Toshiyuki Nakagaki
1,6
*
Transport networks are ubiquitous in both social and biological systems. Robust network performance
involves a complex trade-off involving cost, transport efficiency, and fault tolerance. Biological
networks have been honed by many cycles of evolutionary selection pressure and are likely to yield
reasonable solutions to such combinatorial optimization problems. Furthermore, they develop without
centralized control and may represent a readily scalable solution for growing networks in general. We
show that the slime mold Physarum polycephalum formsnetworkswithcomparableefficiency,fault
tolerance, and cost to those of real-world infrastructure networks—in this case, the Tokyo rail system.
The core mechanisms needed for adaptive network formation can be captured in a biologically
inspired mathematical model that may be useful to guide network construction in other domains.
Transport networks are a critical part of the
infrastructure needed to operate a modern
industrial society and facilitate efficient
movement of people, resources, energy, and
information. Despite their importance, most net-
works have emerged without clear global design
principles and are constrained by the priorities
imposed at their initiation. Thus, the main motiva-
tion historically was to achieve high transport
efficiency at reasonable cost, but with correspond-
ingly less emphasis on making systems tolerant to
interruption or failure. Introducing robustness
inevitably requires additional redundant pathways
that are not cost-effective in the short term. In recent
years, the spectacular failure of key infrastructure
such as power grids (1,2), financial systems (3,4),
airline baggage-handling systems (5), and railway
networks (6), as well as the predicted vulnerability o f
systems such as information networks (7)orsupply
networks (8) to attack, have highlighted the need to
develop networks with greater intrinsic resilience.
Some organisms grow in the form of an inter-
connected network as part of their normal forag-
ing strategy to discover and exploit new resources
(9–12). Such systems continuously adapt to their
environment and must balance the cost of produc-
ing an efficient network with the consequences of
even limited failure in a competitive world. Unlike
anthropogenic infrastructure systems, these biolog-
ical networks have been subjected to successive
rounds of evolutionary selection and are likely to
have reached a point at which cost, efficiency, and
resilience are appropriately balanced. Drawing in-
spiration from biology has led to useful approaches
to problem-solving such as neural networks, ge-
netic algorithms, and efficient search routines de-
veloped from ant colony optimization algorithms
(13). We exploited the slime mold Physarum
polycephalum to develop a biologically inspired
model for adaptive network development.
Physarum is a large, single-celled amoeboid
organism that forages for patchily distributed
food sources. The individual plasmodium ini-
tially explores with a relatively contiguous for-
aging margin to maximize the area searched.
However, behind the margin, this is resolved into
a tubular network linking the discovered food
sources through direct connections, additional in-
termediate junctions (Steiner points) that reduce
the overall length of the connecting network,
and the formation of occasional cross-links that
improve overall transport efficiency and resil-
ience (11,12). The growth of the plasmodium is
influenced by the characteristics of the sub-
strate (14) and can be constrained by physical
barriers (15) or influenced by the light regime
(16), facilitating experimental investigation of
the rules underlying network formation. Thus,
for example, Physarum can find the shortest
path through a maze (15–17) or connect dif-
ferent arrays of food sources in an efficient
manner with low total length (TL) yet short
average minimum distance (MD) between pairs
of food sources (FSs), with a high degree of
fault tolerance (FT) to accidental disconnection
(11,18,19). Capturing the essence of this sys-
tem in simple rules might be useful in guiding
the development of decentralized networks in
other domains.
We ob s erv e d Physarum connecting a template
of 36 FSs that represented geographical locations
of cities in the Tokyo area, and compared the result
with the actual rail network in Japan. The
Physarum plasmodium was allowed to grow from
Tokyo and initially filled much of the available
land space, but then concentrated on FSs by
thinning out the network to leave a subset of larger,
interconnecting tubes (Fig. 1). An alternative
protocol, in which the plasmodium was allowed
to extend fully in the available space and the FSs
were then presented simultaneously, yielded sim-
ilar results. To complete the network formation, we
allowed any excess volume of plasmodium to
1
Research Institute for Electronic Science, Hokkaido University,
Sapporo 060-0812, Japan.
2
PRESTO, JST, 4-1-8 Honcho,
Kawaguchi, Saitama, Japan.
3
Graduate School of Engineering,
Hokkaido University, Sapporo 060-8628, Japan.
4
Department of
Plant Sciences, University of Oxford, Oxford OX1 3RB, UK.
5
Department of Mathematical and Life Sciences, Hiroshima
University, Higashi-Hiroshima 739-8526, Japan.
6
JST, CREST, 5
Sanbancho, Chiyoda-ku, Tokyo, 102-0075, Japan.
*To whom correspondence should be addressed. E-mail:
nakagaki@es.hokudai.ac.jp
www.sciencemag.org SCIENCE VOL 327 22 JANUARY 2010 439
REPORTS
on January 26, 2010 www.sciencemag.orgDownloaded from

accumulate on a large FS outside the arena (LFS
in Fig. 2A).
A range of network solutions were apparent
in replicate experiments (compare Fig. 2A with
Fig. 1F); nonetheless, the topology of many
Physarum networks bore similarity to the real rail
network (Fig. 2D). Some of the differences may
relate to geographical features that constrain the rail
network, such as mountainous terrain or lakes.
These constraints were imposed on the Physarum
network by varying the intensity of illumination, as
the plasmodium avoids bright light (16). This
yielded networks (Fig. 2, B and C) with greater
visual congruence to the real rail network (Fig. 2D).
Networks were also compared with the minimal
spanning tree (MST, Fig. 2E), which is the shortest
possible network connecting all the city positions,
and various derivatives with increasing numbers of
cross-links added (e.g., Fig. 2F), culminating in a
fully connected Delaunay triangulation, which rep-
resents the maximally connected network linking
all the cities.
The performance of each network was char-
acterized by the cost (TL), transport efficiency
(MD), and robustness (FT), normalized to the
corresponding value for the MST to give TL
MST
,
MD
MST
,andFT
MST
. The TL of the Tokyo rail
network was greater than the MST by a factor
of ~1.8 (i.e., TL
MST
≈1.8), whereas the average
TL
MST
for Physarum was 1.75 T0.30 (n= 21).
Illuminated networks gave slightly better clus-
tering around the value for the rail network (Fig.
3A). For comparison, the Delaunay triangulation
was longer than the MST by a factor of ~ 4.6.
Thus, the cost of the solutions found by Physarum
closely matched that of the rail network, with
about 30% of the maximum possible number of
links in place. The transport performance of the
two networks was also similar, with MD
MST
of
0.85 and 0.85 T0.04 for the rail network and the
Physarum networks, respectively. However, the
Physarum networks achieved this with margin-
ally lower overall cost (Fig. 3A).
The converse was true for the fault tolerance
(FT
MST
) in which the real rail network showed
marginally better resilience, close to the lowest
level needed to give maximum tolerance to a single
random failure. Thus, only 4% of faults in the rail
networkwouldleadtoisolationofanypart,
whereas 14 T4% would disconnect the illuminated
Physarum networks, and 20 T13% would
disconnect the unconstrained Physarum networks.
In contrast, simply adding additional links to the
MST to improve network performance resulted
in networks with poor fault tolerance (Fig. 3B).
The trade-off between fault tolerance and cost
was captured in a single benefit-cost measure, ex-
pressed as the ratio of FT/TL
MST
=a. In general,
the Physarum networks and the rail network had
a benefit/cost ratio of ~0.5 for any given TL
MST
(Fig. 3B). The relationship between different a
values and transport efficiency (Fig. 3C) high-
lighted the similarity in aggregate behavior of the
Physarum network when considering all three per-
formance measures (MD
MST
,TL
MST
,andFT
MST
).
Fig. 1. Network formation in Physa-
rum polycephalum.(A)Att=0,a
small plasmodium of Physarum was
placed at the location of Tokyo in an
experimental arena bounded by the
Pacific coastline (white border) and
supplemented with additional food
sourcesateachofthemajorcitiesin
the region (white dots). The horizontal
widthofeachpanelis17cm.(Bto F)
The plasmodium grew out from the
initial food source with a contiguous
margin and progressively colonized
each of the food sources. Behind the
growing margin, the spreading myce-
lium resolved into a network of tubes
interconnecting the food sources.
A
0 hr
D
11 hr
B
5 hr
E
16 hr
8 hr
CF
26 hr
Fig. 2. Comparison of the Physarum
networks with the Tokyo rail network.
(A) In the absence of illumination, the
Physarum network resulted from even
exploration of the available space. (B)
Geographical constraints were imposed
on the developing Physarum network
by means of an illumination mask to
restrict growth to more shaded areas
corresponding to low-altitude regions.
The ocean and inland lakes were also
given strong illumination to prevent
growth. (Cand D) The resulting network
(C) was compared with the rail network
in the Tokyo area (D). (Eand F)The
minimum spanning tree (MST) con-
necting the same set of city nodes (E)
and a model network constructed by
adding additional links to the MST (F).
C
A
D
E
LFS
B
F
22 JANUARY 2010 VOL 327 SCIENCE www.sciencemag.org440
REPORTS
on January 26, 2010 www.sciencemag.orgDownloaded from

The rail network was embedded in the cluster of
results for the Physarum networks with a margin-
ally higher avalue for the same transport effi-
ciency (Fig. 3C).
Overall, we conclude that the Physarum net-
works showed characteristics similar to those of
the rail network in terms of cost, transport efficien-
cy, and fault tolerance. However, the Physarum
networks self-organized without centralized con-
trol or explicit global information by a process of
selective reinforcement of preferred routes and
simultaneous removal of redundant connections.
We developed a mathematical model for adapt-
ive network construction to emulate this behavior,
based on feedback loops between the thickness of
each tube and internal protoplasmic flow (18–22)
in which high rates of streaming stimulate an in-
crease in tube diameter, whereas tubes tend to de-
cline at low flow rates (23). The initial shape of a
plasmodium is represented by a randomly meshed
lattice with a relatively fine spacing, as shown in
Fig. 4 (t= 0). The edges represent plasmodial
tubes in which protoplasm flows, and nodes are
junctions between tubes. Suppose that the pres-
sures at nodes iand jare p
i
and p
j
, respectively,
and that the two nodes are connected by a cyl-
inderoflengthL
ij
and radius r
ij
. Assuming that
flow is laminar and follows the Hagen-Poiseuille
equation, the flux through the tube is
Qij ¼
r4ðpi−pjÞ
8hLij
¼Dijðpi−pjÞ
Lij
ð1Þ
where his the viscosity of the fluid, and D
ij
=
pr
4
/8his a measure of the conductivity of the
tube. As the length L
ij
is a constant, the behavior
of the network is described by the conductivities,
D
ij
, of the edges.
At each time step, a random FS (node 1) is
selected to drive flow through the network, so the
flux includes a source term S
j
Q
1j
=I
0
. A second
random FS is chosen as a sink (node 2) with a
corresponding withdrawal of I
0
such that S
j
Q
2j
=
–I
0
. As the amount of fluid must be conserved,
the inflow and outflow at each internal node must
balance so that i(i≠1, 2), S
j
Q
ij
= 0. Thus, for a
given set of conductivities and selected source
and sink nodes, the flux through each of the
network edges can be computed.
To accommodate the adaptive behavior of the
plasmodium, the conductivity of each tube evolves
according to dD
ij
/dt =f(|Q
ij
|) –D
ij
.Thefirstterm
on the right side describes the expansion of tubes in
response to the flux. The second term represents
the rate of tube constriction, so that in the absence
of flow the tubes will gradually disappear. The
functional form f(|Q|) is given by f(|Q|) = |Q|
g
/(1 +
|Q|
g
), which describes a sigmoidal response where g
is a parameter that controls the nonlinearity of feed-
back (g> 0). A typical simulation result with I
0
=2
and g= 1.8 (Fig. 4) gave a network with features
similar to those of both the Physarum system and
the rail network (Fig. 2, C and D, respectively).
In general, increasing I
0
promoted the for-
mation of alternative routes that improved per-
formance by reducing MD
MST
and made the
network more fault-tolerant, but with increased
cost (Fig. 3, A to C, and fig. S1I). Low values of g
also gave a greater degree of cross-linking with
an increased number of Steiner points (fig. S2, A
and B). Conversely, decreasing I
0
(fig. S1A) or
increasing g(fig. S2I) drove the system toward a
low-cost MST (Fig. 2E), but with an inevitable
decrease in resilience (Fig. 3B). The final net-
work solution also depended slightly on the
stochastic variation assigned to the starting values
of D
ij
. Judicious selection of specific parameter
combinations (I
0
= 0.20, g= 1.15) yielded net-
works with remarkably similar topology and
metrics to the Tokyo rail network (fig. S2B). How-
ever, by increasing I
0
to 2 and gto 1.8, the simula-
tion model also achieved a benefit/cost ratio (a=
FT/TL
MST
) that was better than those of the rail or
Physarum networks, reaching a value of 0.7 with
an almost identical transport efficiency of 0.85
(Fig. 3C). Conversely, the consequence of the in-
creased TL
MST
observed in the rail or Physarum
networks would be to confer greater resilience to
Fig. 3. Transport performance,
resilience, and cost for Physa-
rum networks, model simula-
tions, and the real rail networks.
(A) Transport performance of
each network, measured as the
minimum distance between all
pairs of nodes, normalized to
the MST (MD
MST
)andplotted
against the total length of the
network normalized by the MST
(TL
MST
)asameasureofcost.
Black circles and blue squares
represent results obtained from
Physarum in the absence or
presence of illumination, respectively. The green triangle represents the actual
rail network. Open red circles represent simulation results as I
0
was varied from
0.20 to 7.19 at a fixed g( = 1.80) and initial random fluctuations of D
ij
.(B)Fault
tolerance (FT), measured as the probability of disconnecting part of the network
with failure of a single link. Crosses represent results for reference networks; other
symbols as in (A). Different values of the benefit/cost ratio, a=FT/TL
MST
,are
shown as dashed lines. (C) Relationship between MD
MST
and a. Although the
overall performance of the experiment and that of the real rail network are
clustered together, the simulation model achieves better fault tolerance for the
same transport efficiency.
BC
0.75
0.8
0.85
0.9
0.95
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Performance (MDMST)
0.75
0.8
0.85
0.9
0.95
1
1.0 1.5 2.0 2.5 3.0
0.3
0.6
0.7
α=0.2
0.4
0
0.2
0.4
0.6
0.8
1
Fault tolerance (FT)
1.0 1.5 2.0 2.5 3.0
Performance (MDMST)
Cost (TLMST)Cost (TLMST)Efficiency (FT / TLMST)
A
Fig. 4. Network dynamics for the
simulation model. In this typical time
course for evolution of the simula-
tion, time (t)isshowninarbitrary
units;citiesarebluedots.Eachcity
was modeled as a single FS, apart
from Tokyo, which was an aggregate
of seven FSs to match the importance
of Tokyo as the center of the region.
At the start (t= 0), the available
space was populated with a finely
meshed network of thin tubes. Over
time, many of these tubes died out,
whilst a limited number of tubes be-
came selectively thickened to yield
a stable, self-organized solution. g=
1.80, I
0
=2.00.
t=0
t=1000
t=3000
t=29950
www.sciencemag.org SCIENCE VOL 327 22 JANUARY 2010 441
REPORTS
on January 26, 2010 www.sciencemag.orgDownloaded from

multiple simultaneous failures at the expense of
increased cost, rather than tolerance to a single
disconnection that is evaluated by FT
MST
.
Our biologically inspired mathematical model
can capture the basic dynamics of network
adaptability through iteration of local rules and
produces solutions with properties comparable to
or better than those of real-world infrastructure
networks. Furthermore, the model has a number
of tunable parameters that allow adjustment of
the benefit/cost ratio to increase specific features,
such as fault tolerance or transport efficiency, while
keeping costs low. Such a model may provide a
useful starting point to improve routing protocols
and topology control for self-organized networks
such as remote sensor arrays, mobile ad hoc net-
works, or wireless mesh networks (24).
References and Notes
1. R. Albert, I. Albert, G. Nakarado, Phys. Rev. E 69,
025103R (2004).
2. R. V. Solé, M. Rosas-Casals, B. Corominas-Murtra,
S. Valverde, Phys. Rev. E 77, 026102 (2008).
3. R. M. May, S. Levin, G. Sugihara, Nature 451, 893 (2008).
4. J. Kambhu, S. Weidman, N. Krishnan, Econ. Policy Rev.
13, 1 (2007).
5. House of Commons Transport Committee, The Opening of
Heathrow Terminal 5 HC 543 (Stationery Office, London,
2008).
6. Train Derailment at Hatfield (Independent Investigation
Board, Office of Rail Regulation, London, 2006).
7. R. Albert, H. Jeong, A.-L. Barabási, Nature 406, 378 (2000).
8. R. Carvalho et al., http://arxiv.org/abs/0903.0195 (2009).
9. D. Bebber, J. Hynes, P. Darrah, L. Boddy, M. Fricker,
Proc. R. Soc. London Ser. B 274, 2307 (2007).
10. J. Buhl et al., Behav. Ecol. Sociobiol. 63, 451 (2009).
11. T. Nakagaki, H. Yamada, M. Hara, Biophys. Chem. 107,
1 (2004).
12. T. Nakagaki, R. Kobayashi, Y. Nishiura, T. Ueda,
Proc. R. Soc. London Ser. B 271, 2305 (2004).
13. A. Colorni et al., Int. Trans. Oper. Res. 3, 1 (1996).
14. A. Takamatsu, E. Takaba, G. Takizawa, J. Theor. Biol. 256,
29 (2009).
15. T. Nakagaki, H. Yamada, Á. Tóth, Nature 407, 470 (2000).
16. T. Nakagaki et al., Phys. Rev. Lett. 99, 068104 (2007).
17. T. Nakagaki, H. Yamada, Á. Tóth, Biophys. Chem. 92,47
(2001).
18. A. Tero, K. Yumiki, R. Kobayashi, T. Saigusa, T. Nakagaki,
Theory Biosci. 127, 89 (2008).
19. T. Nakagaki, R. Guy, Soft Matter 4, 57 (2008).
20. T. Nakagaki, T. Saigusa, A. Tero, R. Kobayashi, in
Topological Aspects of Critical Systems and Networks:
Proceedings of the International Symposium,K.Yakubo
et al., Eds. (World Scientific, Singapore, 2007), pp. 94–100.
21. A. Tero, R. Kobayashi, T. Nakagaki, J. Theor. Biol. 244,
553 (2007).
22. A. Tero, R. Kobayashi, T. Nakagaki, Physica A 363, 115
(2006).
23. T. Nakagaki, H. Yamada, T. Ueda, Biophys. Chem. 84,
195 (2000).
24. I. Akyildiz, X. Wang, W. Wang, Comput. Netw. 47, 445
(2005).
25. Supported by MEXT KAKENHI grants 18650054 and
20300105, Human Frontier Science Program grant
RGP51/2007, EU Framework 6 contract 12999 (NEST),
and NERC grant A/S/882.
Supporting Online Material
www.sciencemag.org/cgi/content/full/327/5964/439/DC1
Figs. S1 and S2
17 June 2009; accepted 20 November 2009
10.1126/science.1177894
Measurement of Universal
Thermodynamic Functions for a
Unitary Fermi Gas
Munekazu Horikoshi,
1
*Shuta Nakajima,
2
Masahito Ueda,
1,2
Takashi Mukaiyama
1,3
Thermodynamic properties of matter generally depend on the details of interactions between its
constituent parts. However, in a unitary Fermi gas where the scattering length diverges,
thermodynamics is determined through universal functions that depend only on the particle
density and temperature. By using only the general form of the equation of state and the
equation of force balance, we measured the local internal energy of the trapped gas as a
function of these parameters. Other universal functions, such as those corresponding to the
Helmholtz free energy, chemical potential, and entropy, were calculated through general
thermodynamic relations. The critical parameters were also determined at the superfluid
transition temperature. These results apply to all strongly interacting fermionic systems,
including neutron stars and nuclear matter.
Degenerate two-component Fermi systems
with large scattering lengths are of great
interest in diverse settings such as neutron
stars (1–3), quark-gluon plasma (4), high critical
temperature (T
c
) superconductors (5), and reso-
nantly interacting cold Fermi gases near Feshbach
resonances (6–18). Even though the temperature
of these systems ranges widely from 10
−7
Kfor
cold atoms to more than 10
12
K for quark-gluon
plasma, they exhibit remarkably similar behav-
ior at the unitarity limit. As the scattering length
diverges, the universal thermodynamics that de-
scribes these systems depends only on the particle
density, n, and temperature, T. This assumption is
referred to as the “universal hypothesis (UH)”
(19,20).
In the context of cold atoms, two fermionic
alkali elements,
6
Li and
40
K, have been suc-
cessfully used to explore the physics of the uni-
tarity limit (6–18). This was possible because
of the tunability of the fermion-fermion interac-
tion and the stability of ultracold fermionic
gases near Feshbach resonances (21,22).
Recently, a comparison of the entropy-energy
relations extracted from experimental measure-
ments on both
6
Li and
40
K provided evidence of
universal thermodynamics at the unitarity limit
(23). However, because a unitary Fermi gas is
realized in a harmonic trap, the inhomogeneous
atomic density distribution causes the thermo-
dynamic quantities to be position-dependent.
Therefore, integration over the entire cloud pro-
vides only indirect information on the relation-
ship between each individual thermodynamic
quantity and the particle density. To determine
the universal thermodynamic functions using such
an inhomogeneous system, the thermodynamic
1
Japan Science and Technology Agency, Exploratory Research for
Advanced Technology (ERATO), Macroscopic Quantum Control
Project, 2-11-16 Yayoi, Bunkyo-ku, Tokyo 113-8656, Japan.
2
Department of Physics, University of Tokyo, 7-3-1 Hongo,
Bunkyo-ku, Tokyo 113-0033, Japan.
3
Center for Frontier Science
and Engineering, University of Electro-Communications, 1-5-1
Chofugaoka, Chofu, Tokyo 182-8585, Japan.
*To whom correspondence should be addressed. E-mail:
hori@sogo.t.u-tokyo.ac.jp
Fig. 1. Universal function of the internal en-
ergy. Universal functions of the internal energy
(f
E
[q]=E/Ne
F
) plotted for an ideal Fermi gas
(green diamonds) and for a unitary Fermi gas
(red circles). The data are averaged over a suit-
able temperature range. The error bars show
the data spread of one standard deviation
originating mainly from statistical errors. The
green dashed curve shows the theoretical uni-
versal function for the ideal Fermi gas, whereas
the red solid curve shows the measured univer-
sal function for the unitary Fermi gas. The red
solid curve is obtained by fitting the data repre-
sented by red circles so that it levels off at f
E
[0] =
3(1 + b)/5 = 0.25 at the low-temperature limit,
where bis the universal parameter (15), and ap-
proaches the theoretical value obtained at the
high-temperature limit (20). The blue square cor-
responds to the critical point.
22 JANUARY 2010 VOL 327 SCIENCE www.sciencemag.org
442
REPORTS
on January 26, 2010 www.sciencemag.orgDownloaded from