Research

Entropy Driven Phase Transitions

Entropy-driven phase transitions are interesting and counter-intuitive as they lead to increase in visible order accompanied by a gain in entropy. This is counter to the usual notion that higher the entropy, higher the disorder. Minimal models for studying such transitions are those where the constituent particles interact only through excluded volume interactions. Effectively, (i) no two particles may overlap and (ii) all valid configurations have equal energy. Well-known examples are the density-driven freezing transition in a collection of hard spheres, and the isotropic-nematic phase transition in a collections of hard rods. When the particles are constrained to be on lattices, then the models are called hard core lattice gas models.

The only two control parameters are the shape of the particles and the density. Questions of interest:

• Given the shape of particles, what are the possible phases?

• In what order do they appear with increasing density?

• What is the nature of the phase transitions, if any?

• What is the role of polydispersity?

Though the study of hard core lattice gas models has a long history, the phenomenology of systems of particles with large excluded volume has been difficult to determine due to long-lived metastable states preventing equilibration of the system in numerical simulations. We have been able to conceptualise and implement a Monte Carlo algorithm with cluster moves that overcomes this difficulty. This has allowed us to determine the phase diagram of systems of rods, rectangles, squares, discs, y-shaped molecules etc., numerically, which was not possible earlier, and also provide a theoretical understanding for most of the numerical results.

Related papers:

• A. Shah, D. Dhar and R. Rajesh, The phase transition from nematic to high-density disordered phase in a system of hard rods on a lattice, arXiv:2109.07881

• D. Mandal, G. Rakala, K. Damle, D. Dhar and R. Rajesh, Phases of the hard-plate lattice gas on a three-dimensional cubic lattice, arXiv:2109.02611

• G. Rakala, D. Mandal, S. Biswas, K. Damle, D. Dhar and R. Rajesh, Spontaneous layering and power-law order in the three-dimensional fully-packed hard-plate lattice gas, arXiv:2109.02619

• A. A. A. Jaleel, D. Mandal and R. Rajesh, Hard core lattice gas with third next-nearest neighbor exclusion on triangular lattice: one or two phase transitions?, Journal of Chemical Physics 155, 224101 (2021)

• A. A. A. Jaleel, J. E. Thomas, D. Mandal, Sumedha and R. Rajesh, Rejection free cluster Wang Landau algorithm for hard core lattice gases, arXiv:2108.01402

• D. Dhar and R. Rajesh, Entropy of fully packed hard rigid rods on d-dimensional hyper-cubic lattices, Physical Review E 103, 042130 (2021)

• N. Vigneshwar, D. Mandal, K. Damle, D. Dhar and R. Rajesh, Phase diagram of a system of hard cubes on the cubic lattice, Physical Review E 99, 052129 (2019)

• D. Mandal, T. Nath and R. Rajesh, Phase transitions in a system of hard Y-shaped particles on the triangular lattice, Physical Review E 97, 032131 (2018)

• S. Patra, D. Das, R. Rajesh and M. K. Mitra, Diffusion dynamics and steady states of systems of hard rods on the square lattice, Physical Review E 97, 022108 (2018)

• N. Vigneshwar, D. Dhar and R. Rajesh, Different phases of a system of hard rods on three dimensional cubic lattice, Journal of Statistical Mechanics 2017, 113304 (2017)

• D. Mandal and R. Rajesh, The columnar-disorder phase boundary in a mixture of hard squares and dimers, Physical Review E 96, 012140 (2017)

• D. Mandal, T. Nath and R. Rajesh, Estimating the Critical Parameters of the Hard Square Lattice Gas Model, Journal of Statistical Mechanics 2017, 043201 (2017)

• T. Nath and R. Rajesh, The High Density Phase of the k-NN Hard Core Lattice Gas Model, Journal of Statistical Mechanics 2016, 073203 (2016)

• T. Nath, D. Dhar and R. Rajesh, Stability of columnar order in assemblies of hard rectangles or squares, Europhysics Letters 114, 10003 (2016)

• J. Kundu, J. Stilck and R. Rajesh, Phase diagram of a bidispersed hard rod lattice gas in two dimensions, Europhysics Letters 112, 66002 (2016)

• J. Kundu and R. Rajesh, Phase transitions in systems of hard rectangles with non-integer aspect ratio, European Physical Journal B 88, 133 (2015)

• T. Nath, J. Kundu and R. Rajesh, High-Activity Expansion for the Columnar Phase of the Hard Rectangle Gas, Journal of Statistical Physics 160, 1173 (2015)

• J. Stilck and R. Rajesh, Polydispersed rods on the square lattice, Physical Review E 91, 012106 (2015)

• J. Kundu and R. Rajesh, Asymptotic Behavior of the Isotropic-Nematic and Nematic-Columnar Phase Boundaries for the System of Hard Rectangles on a Square lattice, Physical Review E 91, 012105 (2015)

• T. Nath and R. Rajesh, Multiple Phase Transitions in Extended Hard Core Lattice Gas Models in Two Dimensions, Physical Review E 90, 012120 (2014)

• J. Kundu and R. Rajesh, Phase transitions in a system of hard rectangles on the square lattice, Physical Review E 89, 052124 (2014)

• J. Kundu and R. Rajesh, Reentrant disordered phase in a system of repulsive rods on a Bethe-like lattice, Physical Review E 88, 012134 (2013)

• J. Kundu, R. Rajesh, D. Dhar and J. Stilck, The nematic-disordered phase transition in systems of long rigid rods on two dimensional lattices, Physical Review E 87, 032103 (2013)

• J. Kundu, R. Rajesh, D. Dhar and J. Stilck, A Monte Carlo algorithm for studying phase transition in systems of hard rigid rods, AIP Conf. Proc. 1447, 113 (2012)

• D. Dhar, R. Rajesh and J. Stilck, Hard rigid rods on a Bethe-like lattice, Physical Review E 84, 011140 (2011)

Granular Systems

Granular materials are ubiquitous in nature. Examples include geophysical flows, large-scale structure formation of the universe, sand dunes, craters, steel balls, etc. Granular gases, due to inelastic collisions between particles, behave remarkably different from their elastic counterparts. In particular, in the absence of external driving, inelasticity induces clustering, leading to the system becoming inhomogeneous. In the presence of driving, the system is driven to a steady state that has very different properties from a corresponding elastic gas of the same mean energy. For example, the velocity distribution is no longer Maxwellian.

We have studied in detail dilute granular gases, both freely cooling as well as driven. Questions of interest:

• What are the tails of the velocity distribution of a driven granular gas?

• What are the large scale temporal and spatial properties of a cooling granular gas?

• What are the scaling laws describing a granular system subjected to a sudden localised perturbation?

Related papers:

• A. Biswas, V. V. Prasad and R. Rajesh, Mpemba effect in anisotropically driven inelastic Maxwell gases, arXiv:2105.01972

• A. Biswas, V. V. Prasad and R. Rajesh, Mpemba effect in an anisotropically driven granular gas, arXiv:2104.08730

• A. Biswas, V. V. Prasad, O. Raz and R. Rajesh, Mpemba effect in driven granular Maxwell gases, Physical Review E 102, 012906 (2020)

• A. Biswas, V. V. Prasad and R. Rajesh, Asymptotic velocity distribution of a driven one dimensional binary granular Maxwell gas, Journal of Statistical Mechanics 013202 (2020)

• V. V. Prasad, D. Das, S. Sabhapandit and R. Rajesh, Steady state velocity distribution of driven granular gases, Journal of Statistical Mechanics 2019, 063201 (2019)

• V. V. Prasad and R. Rajesh, Asymptotic Behavior of the Velocity Distribution of Driven Inelastic Gas with Scalar Velocities: Analytical Results, Journal of Statistical Physics 176, 1409 (2019)

• J. P. Joy, S. N. Pathak, D. Das and R. Rajesh, Shock propagation in locally driven granular systems, Physical Review E 96, 032908 (2017)

• V. V. Prasad, D. Das, S. Sabhapandit and R. Rajesh, Velocity distribution of driven inelastic one-component Maxwell gas, Physical Review E 93, 032909 (2017)

• S. N. Pathak, D. Das and R. Rajesh, Inhomogeneous Cooling of the Rough Granular Gas in Two Dimensions, Europhysics Letters 107, 44001 (2014)

• S. N. Pathak, Z. Jabeen, D. Das and R. Rajesh, Energy decay in three-dimensional freely cooling granular gas, Physical Review Letters 112, 038001 (2014)

• S. N. Pathak, Z. Jabeen, R. Rajesh and P. Ray, Shock propagation in a visco-elastic granular gas, AIP Conf. Proc. 1447, 193 (2012)

• S. N. Pathak, Z. Jabeen, P. Ray and R. Rajesh, Shock propagation in granular flow subjected to an external impact, Physical Review E 85, 061301 (2012)

• S. Dey, D. Das and R. Rajesh, Lattice models for ballistic aggregation in one-dimension, Europhysics Letters 93, 44001 (2011)

• M. Shinde, D. Das and R. Rajesh, Coarse grained dynamics of the freely cooling granular gas in one dimension, Physical Review E 84, 031310 (2011)

• Z. Jabeen, R. Rajesh and P. Ray, Universal scaling dynamics in a perturbed granular gas, Europhysics Letters 89, 34001 (2010)

• M. Shinde, D. Das and R. Rajesh, Equivalence of the freely cooling granular gas to the sticky gas, Physical Review E 79, 021303 (2009)

• M. Shinde, D. Das and R. Rajesh, Violation of Porod law in a freely cooling granular gas in one dimension, Physical Review Letters 99, 234505 (2007)

Mechanics of Bone/Composites

Cortical or compact bone, found in the midshaft of load-bearing bones such as femurs and tibiae, is a brittle, porous biomaterial. Being a living tissue, the local microstructure and porosity network of the cortical bone evolves in response to the mechanical stresses to which the bone is subjected, and this in turn modifies the local mechanical properties. Understanding the relationship between microstructure and mechanical properties is crucial for applications such as the extraction of bone grafts, in designing mechanically compatible implants and porous scaffolds for bone tissue engineering, in order to interpret loading history, evaluate the effectiveness of chemical and physical therapeutical measures for bone healing etc. An important aspect of this understanding is the development and testing of models that incorporate microstructural features and predict material properties such as failure strength, elastic modulus, fracture paths, etc. Such models, if general enough, would also be of use in understanding failure behaviour of a wider class of brittle materials with a well-defined porosity network, such as wood, rock, etc.

Questions of interest:

• What is the correlation between the microstructure and the nature of local stresses?

• How does the microstructure help in withstanding the applied stresses?

• Can a predictive microscopic model be made that is able to reproduce the macroscopic response of the bone to externally applied loading?

Related papers:

• D. Kumar, A. Banerjee and R. Rajesh, Crushing of square honeycombs using disordered spring network model, Mechanics of Materials, 160, 103947 (2021) [journal link]

• R. P. S. Parihar, D. V. Mani, A. Banerjee and R. Rajesh, Role of spatial patterns in fracture of disordered multiphase materials, Physical Review E 102, 053002 (2020)

• A. Mayya, A. Banerjee and R. Rajesh, Role of porosity and matrix behavior on compressive fracture of Haversian bone using random spring network model, Journal of the Mechanical Behavior of Biomedical Materials 83, 108 (2018)

• A. Mayya, A. Banerjee and R. Rajesh, On role of matrix behavior in compressive fracture of bovine cortical bone, Physical Review E 96, 053001 (2017)

• A. Mayya, P. Praveen, A. Banerjee and R. Rajesh, Splitting fracture in bovine bone using a porosity based spring network model, Journal of The Royal Society Interface 13, 20160809 (2016)

• A. Mayya, A. Banerjee and R. Rajesh, Haversian microstructure in bovine femoral cortices: an adaptation for improved compressive strength, Material Science and Engineering C 59, 454 (2016)

• A. Mayya, A. Banerjee and R. Rajesh, Mammalian cortical bone in tension is non-Haversian, Scientific Reports 3, 2533 (2013)

Shock Propagation

When a large amount of energy is released in a localized region, the perurbed matter may move faster than the rate at which energy is trabsferred using heat or sound modes, resulting in a shock wave. We have focussed on determining the scaling behaviour of different thermodynamic quantities, in shock like conditions, in elastic media (like a dilute gas) as well as inelastic media (granular material).

Questions of interest:

• Can the scaling laws seen in experiements on granular systems be obtained using simple models of inealstic spheres?

• Are the predictions of the hydrodynamic theories for thermodynamic quantities consistent with molecular dynamics simulations of hard spheres?

Related papers:

• A. Kumar and R. Rajesh, Blast waves in two and three dimensions: Euler versus Navier Stokes equations, arXiv:2111.09213

• J. P. Joy and R. Rajesh, Shock propagation in the hard sphere gas in two dimensions: comparison between simulations and hydrodynamics, Journal of Statistical Physics 184, 3 (2021) [journal link]

• J. P. Joy, S. N. Pathak and R. Rajesh, Shock propagation following an intense explosion: comparison between hydrodynamics and simulations, Journal of Statistical Physics 182, 34 (2021)

• J. P. Joy, S. N. Pathak, D. Das and R. Rajesh, Shock propagation in locally driven granular systems, Physical Review E 96, 032908 (2017)

• S. N. Pathak, Z. Jabeen, R. Rajesh and P. Ray, Shock propagation in a visco-elastic granular gas, AIP Conf. Proc. 1447, 193 (2012)

• S. N. Pathak, Z. Jabeen, P. Ray and R. Rajesh, Shock propagation in granular flow subjected to an external impact, Physical Review E 85, 061301 (2012)

• Z. Jabeen, R. Rajesh and P. Ray, Universal scaling dynamics in a perturbed granular gas, Europhysics Letters 89, 34001 (2010)

Non-Equilibrium Statistical Mechanics

Unlike the case of equilibrium statistical mechanics, there is no general formalism for calculating the expectation values of physical observables in systems far from equilibrium. In the absence of such a formalism, one approach is to look at simple systems that are amenable to exact analysis, yet are relevant for some physical phenomena. We have focussed on interacting particle systems where the basic stochastic processes include aggregation, fragmentation, evaporation, input of particles, etc. Such models are relevant for aggregation of polymers, rings of Saturn, river networks, coarsening, etc.

Questions of interest:

• Can Kolmogorov theory of fluid turbulence and its observed breakdown be worked out for model systems?

• Are there general laws governing systems with widely separated driving and dissipation scales in the presence of a conserved quantity?

• What is the role of nonlinear interactions in systems with a conserved quantity?

• How do systems far from equilibrium coarsen when approaching its ordered steady state?

• What are the universality classes of non-equilibrium phase transitions?

Related papers:

• V. S. Akella, R. Rajesh and M. V. Panchagnula, Levy walking droplets, Physical Review Fluids 5, 084002 (2020)

• R. Dandekar, S. Chakraborti and R. Rajesh, Hard core run and tumble particles on a one dimensional lattice, Physical Review E 102, 062111 (2020)

• C. Connaughton, A. Dutta, R. Rajesh, N. Siddharth and O. Zaboronski, Stationary mass distribution and non-locality in models of coalescence and shattering, Physical Review E 97, 022137 (2018)

• C. Connaughton, A. Dutta, R. Rajesh and O. Zaboronski, Universality properties of steady driven coagulation with collisional evaporation, Europhysics Letters 117, 10002 (2017)

• C. Connaughton, R. Rajesh, R. Tribe and O. Zaboronski, Non-equilibrium phase diagram for a model with coalescence, evaporation and deposition, Journal of Statistical Physics 152, 1115 (2013)

• R. Singh and R. Rajesh, Comment on ``Growth Inside a Corner: The limiting Interface Shape'', Physical Review Letters 109, 259601 (2012)

• R. C. Ball, C. Connaughton, P. P. Jones, R. Rajesh and O. Zaboronski, Collective Oscillations in Irreversible Coagulation Driven by Monomer Inputs and Large-Cluster Outputs, Physical Review Letters 109, 168304 (2012)

• S. Dey, D. Das and R. Rajesh, Spatial structures and giant number fluctuations in models of active matter, Physical Review Letters 108, 238001 (2012)

• C. Connaughton, R. Rajesh and O. Zaboronski, On the Non-equilibrium Phase Transition in Evaporation-Deposition Models, Journal of Statistical Mechanics P09016 (2010)

• R. Rajesh, Multi-avalanche correlations in directed sandpile models, Europhysics Letters, 85, 10001 (2009)

• C. Connaughton, R. Rajesh and O. Zaboronski, Constant flux relation for diffusion limited cluster-cluster aggregation, Physical Review E 78, 041403 (2008)

• C. Connaughton, R. Rajesh and O. Zaboronski, Constant flux relation for aggregation models with desorption and fragmentation, Physica A 384, 108 (2007)

• C. Connaughton, R. Rajesh and O. Zaboronski, Constant flux relation for driven dissipative systems, Physical Review Letters 98, 080601 (2007)

• R. Munasinghe, R. Rajesh, R. Tribe and O. Zaboronski, Multi-scaling of the n-point density function for coalescing brownian motions, Communications in Mathematical Physics 268, 717 (2006)

• C. Connaughton, R. Rajesh and O. Zaboronski, Cluster-Cluster Aggregation as an Analogue of a Turbulent Cascade: Kolmogorov Phenomenology, Scaling Laws and the Breakdown of self-similarity, Physica D 222, 97 (2006)

• R. Munasinghe, R. Rajesh and O. Zaboronski, Multi-Scaling of Correlation Functions in Single Species Reaction-Diffusion Systems, Physical Review E 73, 051103 (2006)

• A. Mahmood and R. Rajesh, Cosmological mass functions and moving barrier models, astro-ph/0502513

• C. Connaughton, R. Rajesh and O. Zaboronski, Breakdown of Kolmogorov scaling in models of cluster aggregation, Physical Review Letters 94, 194503 (2005)

• R. Rajesh and O. Zaboronski, Survival probability of a diffusing test particle in a system of coagulating and annihilating random walkers, Physical Review E 70, 036111 (2004)

• R. Rajesh, Nonequilibirium phase transitions in models of adsorption and desorption, Physical Review E 69, 036128 (2004)

• C. Connaughton, R. Rajesh and O. Zaboronski, Stationary kolmogorov solutions of the smoluchowski aggregation equation with a source term, Physical Review E 69, 061114 (2004)

• S. Krishnamurthy, R. Rajesh and O. Zaboronski, Persistence properties of a system of coagulating and annihilating random walkers, Physical Review E 68, 046103 (2003)

• S. Krishnamurthy, R. Rajesh and O. Zaboronski, Kang-Redner small-mass anomaly in cluster-cluster aggregation, Physical Review E 66, 066118 (2002)

• R. Rajesh, D. Das, B. Chakraborty and M. Barma, Aggregate formation in a system of coagulating and fragmenting particles with mass-dependent diffusion rates, Physical Review E 66, 056104 (2002)

• R. Rajesh and S. Krishnamurthy, Effect of spatial bias on the nonequilibrium phase transition in a model of coagulating and fragmentating particles, Physical Review E 66, 046132 (2002)

• R. Rajesh and S. N. Majumdar, Exact tagged particle correlations in the random average process, Physical Review E 64, 036103 (2001)

• R. Rajesh and S. N. Majumdar, Exact phase diagram of a model with aggregation and chipping, Physical Review E 63, 036114 (2001)

• R. Rajesh and S. N. Majumdar, Exact calculation of the spatiotemporal correlations in the Takayasu model and in the q model of force fluctuations in bead packs, Physical Review E 62, 3186 (2000)

• R. Rajesh and S. N. Majumdar, Conserved mass models and particle systems in one dimension, Journal of Statistical Physics 99, 943 (2000)

• R. Rajesh and D. Dhar, An exactly solvable anisotropic directed percolation model in three dimensions, Physical Review Letters 81, 1646 (1998)

Charged Polymers

In chemical, pharmaceutical, food and bio-industries, applications such as gene therapy, drug coating, water purification, colour removal, paper making etc., involve charged polymers in solution. The mechanical and chemical properties of these polymers depend on their conformational state which could be linear and extended, compact and collapsed or in the form of complex aggregates. It is of importance to determine the precise role of electric charge in determining the conformational properties.

We study the large scale structure of charged polymers in solution (polyelectrolytes) through a combinational of large scale simulations and theoretical modelling. Questions of interest:

• What are the effective interactions driving the collapse of a like-charged polymer?

• What the effects of solvent quality, bending rigidity, dressed counterions on the collapse transition?

• What are the different phases that a collection of charged polymers can exist in?

• What is the dynamics of aggregation for a collection of charged polymers, and how does it depend on rigidity?

Related papers:

• A. M. Tom, R. Rajesh and S. Vemparala, Aggregation of flexible polyelectrolytes: Phase diagram and dynamics, Journal of Chemical Physics 147, 144903 (2017)

• A. M. Tom, S. Vemparala, R. Rajesh and N. V. Brilliantov, Regimes of strong electrostatic collapse of a highly charged polyelectrolyte in a poor solvent, Soft Matter 13, 1862 (2017)

• A. M. Tom, S. Vemparala, R. Rajesh and N. V. Brilliantov, Mechanism of chain collapse of strongly charged polyelectrolytes, Physical Review Letters 117, 147801 (2016)

• A. M. Tom, R. Rajesh and S. Vemparala, Aggregation Dynamics of Rigid Polyelectrolytes, Journal of Chemical Physics 144, 034904 (2016)

• A. Varghese, R. Rajesh and S. Vemparala, Aggregation of rod-like polyelectrolyte chains in the presence of monovalent counterions, Journal of Chemical Physics 137, 234901 (2012)

• A. Varghese, S. Vemparala and R. Rajesh, Ensemble Equivalence for Counterion Condensation on a Two Dimensional Charged Disk, Physical Review E 85, 011119 (2012)

• A. Varghese, S. Vemparala and R. Rajesh, Phase transitions of a single polyelectrolyte chain in a poor solvent with multivalent counterions, AIP Conf. Proc. 1447, 129 (2012)

• A. Varghese, S. Vemparala and R. Rajesh, Phase transitions of a single polyelectrolyte in a poor solvent with explicit counterions, Journal of Chemical Physics 135, 154902 (2011)

Neutral Polymers

In the study of polymers, statistical mechanics plays an important role in understanding their large scale structure, different phases and the nature of the transitions between them. Our research has focused on understanding some of these aspects in different neutral polymer systems.

Questions of interest:

• What are the phases of a pressurised ring polymer?

• What is a phase diagram of a polymer in the presence of an attractive wall?

• What are the force fluctuations in a tethered polymer when stretched?

Related papers:

• A. A. A. Jaleel, M. Ponmurugan, R. Rajesh and S. V. M. Satyanarayana, Phase transitions in a linear self-interacting polymer on FCC lattice using flat energy interacting growth walk algorithm, Journal of Statistical Mechanics 2018, 113301 (2018)

• A. Varghese, S. Vemparala and R. Rajesh, Force fluctuations in stretching a tethered polymer, Physical Review E 88, 022134 (2013)

• M. K. Mitra, G. I. Menon and R. Rajesh, Thermodynamic behaviour of two-dimensional vesicles revisited, European Physical Journal E 35, 30 (2012)

• M. K. Mitra, G. I. Menon and R. Rajesh, Asymptotic behaviour of convex and column-convex lattice polygons with fixed area and varying perimeter, Journal of Statistical Mechanics P07029 (2010)

• R. Rajesh, D. Giri, I. Jensen and S. Kumar, Role of pulling direction in understanding the energy landscape of proteins, Physical Review E 78, 021905 (2008)

• M. K. Mitra, G. I. Menon and R. Rajesh, Asymptotic Behavior of Inflated Lattice Polygons, Journal of Statistical Physics 131, 393 (2008)

• M. K. Mitra, G. I. Menon and R. Rajesh, Phase transitions in pressurised semiflexible polymer rings, Physical Review E 77, 041802 (2008)

• R. Rajesh and D. Dhar, Convex lattice polygons of fixed area with perimeter dependent weights, Physical Review E 71, 016130 (2005)

• R. Rajesh, D. Dhar, D. Giri, S. Kumar and Y. Singh, The adsorption and collapse transitions in a linear polymer chain near an attractive wall, Physical Review E 65, 056124 (2002)

Enumeration Problems

Enumerative combinatorics has been an area of considerable interest in Statistical Mechanics. We have focussed on a few problems using numerical and analytical methods.

Questions of interest:

• What is the asymptotic behaviour of solid partitions?

• What is the asymptotic behaviour of different kinds of polygons whose perimeter and volume are weighted?

Related papers:

• M. K. Mitra, G. I. Menon and R. Rajesh, Asymptotic behaviour of convex and column-convex lattice polygons with fixed area and varying perimeter, Journal of Statistical Mechanics P07029 (2010)

• M. K. Mitra, G. I. Menon and R. Rajesh, Asymptotic Behavior of Inflated Lattice Polygons, Journal of Statistical Physics 131, 393 (2008)

• R. Rajesh and D. Dhar, Convex lattice polygons of fixed area with perimeter dependent weights, Physical Review E 71, 016130 (2005)

• V. Mustonen and R. Rajesh, Numerical estimation of the asymptotic behaviour of solid partitions of an integer, Journal of Physics A 36, 6651 (2003)