{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,15]],"date-time":"2026-03-15T01:24:04Z","timestamp":1773537844276,"version":"3.50.1"},"reference-count":26,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2017,9,26]],"date-time":"2017-09-26T00:00:00Z","timestamp":1506384000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Broadly distributed random variables with a power-law distribution     f  ( m )  \u223c  m  - ( 1 + \u03b1 )       are known to generate condensation effects. This means that, when the exponent    \u03b1    lies in a certain interval, the largest variable in a sum of N (independent and identically distributed) terms is for large N of the same order as the sum itself. In particular, when the distribution has infinite mean (    0 &lt; \u03b1 &lt; 1    ) one finds unconstrained condensation, whereas for     \u03b1 &gt; 1    constrained condensation takes places fixing the total mass to a large enough value     M =  \u2211  i = 1  N   m i  &gt;  M c     . In both cases, a standard indicator of the condensation phenomenon is the participation ratio      Y k  =  \u2329  \u2211 i   m i k  \/   (  \u2211 i   m i  )  k  \u232a      (    k &gt; 1    ), which takes a finite value for     N \u2192 \u221e     when condensation occurs. To better understand the connection between constrained and unconstrained condensation, we study here the situation when the total mass is fixed to a superextensive value     M \u223c  N  1 + \u03b4       (    \u03b4 &gt; 0    ), hence interpolating between the unconstrained condensation case (where the typical value of the total mass scales as     M \u223c  N  1 \/ \u03b1       for     \u03b1 &lt; 1    ) and the extensive constrained mass. In particular we show that for exponents     \u03b1 &lt; 1     a condensate phase for values     \u03b4 &gt;  \u03b4 c  = 1 \/ \u03b1 - 1     is separated from a homogeneous phase at     \u03b4 &lt;  \u03b4 c      from a transition line,     \u03b4 =  \u03b4 c     , where a weak condensation phenomenon takes place. We focus on the evaluation of the participation ratio as a generic indicator of condensation, also recalling or presenting results in the standard cases of unconstrained mass and of fixed extensive mass.<\/jats:p>","DOI":"10.3390\/e19100517","type":"journal-article","created":{"date-parts":[[2017,9,26]],"date-time":"2017-09-26T16:14:45Z","timestamp":1506442485000},"page":"517","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["Participation Ratio for Constraint-Driven Condensation with Superextensive Mass"],"prefix":"10.3390","volume":"19","author":[{"given":"Giacomo","family":"Gradenigo","sequence":"first","affiliation":[{"name":"Laboratoire Interdisciplinaire de Physique (LIPHY), Universit\u00e9 Grenoble Alpes and CNRS, F-38000 Grenoble, France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Eric","family":"Bertin","sequence":"additional","affiliation":[{"name":"Laboratoire Interdisciplinaire de Physique (LIPHY), Universit\u00e9 Grenoble Alpes and CNRS, F-38000 Grenoble, France"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2017,9,26]]},"reference":[{"key":"ref_1","first-page":"024005","article-title":"Conditioned random walks and interaction-driven condensation","volume":"50","author":"Evans","year":"2016","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"M\u00e9zard, M., Parisi, G., and Virasoro, M. (1987). Spin Glass Theory and Beyond, World Scientific.","DOI":"10.1142\/0271"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"7997","DOI":"10.1088\/0305-4470\/30\/23\/004","article-title":"Universality classes for extreme-value statistics","volume":"30","author":"Bouchaud","year":"1997","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"505","DOI":"10.1016\/S0550-3213(97)00192-2","article-title":"Condensation in the Backgammon model","volume":"493","author":"Bialas","year":"1997","journal-title":"Nucl. Phys. B"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"3691","DOI":"10.1103\/PhysRevLett.81.3691","article-title":"Nonequilibrium Phase Transitions in Models of Aggregation, Adsorption, and Dissociation","volume":"81","author":"Majumdar","year":"1998","journal-title":"Phys. Rev. Lett."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"389","DOI":"10.1023\/A:1026008532442","article-title":"Condensation in the zero range process: Stationary and dynamical properties","volume":"113","author":"Grosskinsky","year":"2003","journal-title":"J. Stat. Phys."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"180601","DOI":"10.1103\/PhysRevLett.94.180601","article-title":"Nature of the Condensate in Mass Transport Models","volume":"94","author":"Majumdar","year":"2005","journal-title":"Phys. Rev. Lett."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"357","DOI":"10.1007\/s10955-006-9046-6","article-title":"Canonical Analysis of Condensation in Factorised Steady States","volume":"123","author":"Evans","year":"2006","journal-title":"J. Stat. Phys."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"R195","DOI":"10.1088\/0305-4470\/38\/19\/R01","article-title":"Nonequilibrium statistical mechanics of the Zero-Range Process and related models","volume":"38","author":"Evans","year":"2005","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_10","unstructured":"Jacobsen, J., Ouvry, S., Pasquier, V., Serban, D., and Cugliandolo, L.F. (2008). Real-space Condensation in Stochastic Mass Transport Models. Exact Methods in Low-Dimensional Statistical Physics and Quantum Computing, Oxford University Press. Les Houches Lecture Notes for the Summer School."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"090602","DOI":"10.1103\/PhysRevLett.103.090602","article-title":"Condensation in Temporally Correlated Zero-Range Dynamics","volume":"103","author":"Hirschberg","year":"2009","journal-title":"Phys. Rev. Lett."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Whitehouse, J., Costa, A., Blythe, R.A., and Evans, M.R. (2014). Maintenance of order in a moving strong condensate. J. Stat. Mech., P11029.","DOI":"10.1088\/1742-5468\/2014\/11\/P11029"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"095001","DOI":"10.1088\/1751-8113\/47\/9\/095001","article-title":"Condensation in stochastic mass transport models: Beyond the zero-range process","volume":"47","author":"Evans","year":"2014","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Evans, M.R., and Majumdar, S.N. (2008). Condensation and extreme value statistics. J. Stat. Mech., P05004.","DOI":"10.1088\/1742-5468\/2008\/05\/P05004"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"020602","DOI":"10.1103\/PhysRevLett.112.020602","article-title":"Constraint-Driven Condensation in Large Fluctuations of Linear Statistics","volume":"112","author":"Evans","year":"2014","journal-title":"Phys. Rev. Lett."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"455004","DOI":"10.1088\/1751-8113\/47\/45\/455004","article-title":"Condensation Transition in Joint Large Deviations of Linear Statistics","volume":"47","author":"Evans","year":"2014","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"012143","DOI":"10.1103\/PhysRevE.90.012143","article-title":"Condensation of Fluctuations in and out of Equilibrium","volume":"90","author":"Zannetti","year":"2014","journal-title":"Phys. Rev. E"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1016\/j.jnoncrysol.2014.07.039","article-title":"Singular behavior of fluctuations in a relaxation process","volume":"407","author":"Corberi","year":"2015","journal-title":"J. Non-Cryst. Solids"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"20004","DOI":"10.1209\/0295-5075\/111\/20004","article-title":"The Grand Canonical catastrophe as an istance of condensation of fluctuations","volume":"111","author":"Zannetti","year":"2015","journal-title":"Europhys. Lett."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"052138","DOI":"10.1103\/PhysRevE.95.052138","article-title":"Heat fluctuations of Brownian oscillators in nonstationary processes: Fluctuation theorem and condensation transition","volume":"95","author":"Crisanti","year":"2017","journal-title":"Phys. Rev. E"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"010602","DOI":"10.1103\/PhysRevLett.97.010602","article-title":"Interaction driven real-space condensation","volume":"97","author":"Evans","year":"2006","journal-title":"Phys. Rev. Lett."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"909","DOI":"10.1029\/WR004i005p00909","article-title":"Noah, Joseph, and operational hydrology","volume":"4","author":"Mandelbrot","year":"1968","journal-title":"Water Resour. Res."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1016\/j.physa.2007.08.016","article-title":"Fractional Ornstein-Uhlenbeck processes. Joseph effect in models with infinite variance","volume":"387","author":"Magdziarz","year":"2008","journal-title":"Physica A"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"026128","DOI":"10.1103\/PhysRevE.67.026128","article-title":"Subdiffusion and localization in the one dimensional trap model","volume":"67","author":"Bertin","year":"2003","journal-title":"Phys. Rev. E"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"186","DOI":"10.1016\/S0167-2789(97)00086-9","article-title":"From random walks to spin glasses","volume":"107","author":"Derrida","year":"1997","journal-title":"Physica D"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1023\/A:1009913901219","article-title":"Large deviations of heavy-tailed sums with applications in insurance","volume":"1","author":"Mikosch","year":"1998","journal-title":"Extremes"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/19\/10\/517\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T18:46:01Z","timestamp":1760208361000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/19\/10\/517"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,9,26]]},"references-count":26,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2017,10]]}},"alternative-id":["e19100517"],"URL":"https:\/\/doi.org\/10.3390\/e19100517","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,9,26]]}}}