Research

Nonequilibrium Phenomena

Since its inception, modern condensed matter physics and statistical mechanics has dealt almost exclusively with the study of systems in thermal equilibrium. However, many interesting and qualitatively different phenomena occur only in systems that are not in equilibrium. This is what my research focuses on.

Energy transport in one-dimensional systems

When the opposite ends of an object are kept at different temperatures, a steady heat current flows from the hot side to the cold side. When the temperature difference is small, the heat current obeys “Fourier’s law”, analogous to Ohm’s law, which defines a thermal conductivity. However, in many one-dimensional models, heat transport is anomalous: if the length of the system is increased for a fixed temperature difference, the heat current is not inversely proportional to the length. In other words, the calculated thermal conductivity depends on the length of the system.

An analytical treatment of this problem has been constructed, showing that for systems where interparticle interactions conserve momentum, the thermal conductivity should diverge as the cube root of the length of the system. In numerical simulations, one has to work with extremely large systems before this behavior is seen because of the difficulties in thermalizing one-dimensional systems. We have proposed a model where thermalization is efficient, and the theoretically predicted scaling form is easily seen. We have also verified that the Kubo formula, which is commonly used to calculate thermal conductivity, is very sensitive to boundary conditions for such systems and should not be used with periodic boundary conditions. Experimentally, the most promising candidate to show this behavior is a freely suspended nanotube.

  1. “Green-Kubo formula for heat conduction in open systems”, A. Kundu, A. Dhar and O. Narayan, J. Stat. Mech. L03001 (2009).
  2. “Local temperature and universal heat conduction in FPU chains”, T. Mai, A. Dhar and O. Narayan, Phys. Rev. Lett. 98, 184301 (2007).
  3. “Correlations and scaling in one-dimensional heat conduction”, J.M. Deutsch and O. Narayan, Phys. Rev. E 68, 041203 (2003).
  4. “Anomalous heat conduction in one-dimensional momentum conserving systems,” O. Narayan and S. Ramaswamy, Phys. Rev. Lett. 89, 200601 (2002).

Data traffic networks, etc.

The standard theory of data traffic networks is borrowed from the successful treatment of telephone networks, and assumes that the traffic flowing in the network has only short range temporal correlations. This allows a description in terms of Markov models, for which many powerful results can be proved analytically. However, direct measurements on data traffic in networks reveals long tails in the temporal correlations, which seem to decay as power laws. We demonstrated the importance of these tails in predicting network performance. For a single server with power law correlated Gaussian input traffic, we have analytically obtained an expression for the tail of the resulting queue length distribution. Some other engineering papers are also listed here.

  1. “Scaling of load in communications networks”, O. Narayan and I. Saniee, http://arxiv.org/0906.4138
  2. “Multi-scaling models of TCP/IP and sub-frame VBR video traffic”, A. Erramilli, O. Narayan, A. Neidhardt and I. Saniee, IEEE J. Commun. Netw. 3, 383 (2001).
  3. “Exact asymptotic queue-length distribution for fractional Brownian traffic”, O. Narayan, Adv. Perf. Anal. 1, 39 (1998).
  4. “Experimental queueing analysis with long-range dependent packet traffic”, A. Erramilli, O. Narayan and W. Willinger, IEEE/ACM Trans. Networking 4, 209 (1996).
  5. “Analyzing oscillators using multitime PDEs”, O. Narayan and J.S. Roychowdhury, IEEE Trans. Circuits and Systems I, 50, 894 (2003).
  6. “Accounting for boundary effects in nearest neighbor searching”, S. Arya, D. Mount and O. Narayan, Discrete and Computational Geometry 16, 155 (1996).

Nonreciprocity

A question that is related to heat transport is what happens when the reservoirs at the two ends of the system are not heat baths in thermal equilibrium (although at different temperatures), but energy reservoirs out of equilibrium. We have shown through numerical simulations that if identical non-equilibrium reservoirs are placed at the two ends of the system, a steady state energy flow can be established: the system acts like a “ratchet”. We have also considered the case when the “reservoirs” at the two ends are sources of waves, emitting monochromatic waves. If the system is modeled as a nonlinear medium, nonreciprocal wave transmission is possible: the transmission coefficient is not the same in both directions. We have discovered an exact identity that relates this nonreciprocal transmission to a (apparent, not actual) violation of the second law of thermodynamics, that is valid to all orders in perturbation theory.

  1. “Exact identity for nonlinear wave propagation”, D. Ralph, O. Narayan and R. Montgomery, Phys. Rev. E 77, 056219 (2008).
  2. “Nonreciprocity and the second law of thermodynamics: an exact relation for nonlinear media”, O. Narayan and A. Dhar, Europhysics Lett. 67, 559 (2004).
  3. “Ratchet for energy transport between identical reservoirs”, S. Das, O. Narayan and S. Ramaswamy, Phys. Rev. E 66, RC 050103 (2002).

Hysteresis in magnets

The hysteresis loop for ferromagnets is generally assumed to be a simple closed loop, whose area is a measure of how much the magnetic domains get pinned by local disorder. Surprisingly, we have found that hysteresis loops can have more complicated shapes once frustration is present: multicycles, where the hysteresis loop closes once every two (or several) cycles of the magnetic field can be seen. These can be experimentally observed by cycling the magnetic field and measuring the subharmonic response. We first demonstrated this for small spin glass systems, and then extended the work to ferromagnetic nanopillar arrays (where the dipolar interaction provides frustration). In this process, we also discovered that for systems where the energy is invariant under spin (and magnetic field) reversal, the domain patterns observed when lowering the field from large positive or negative fields need not be complementary.

  1. “Hysteresis multicyles in nanomagnet arrays”, J.M. Deutsch, T. Mai and O. Narayan, Phys. Rev. E 71, 026120 (2005).
  2. “Disorder induced microscopic magnetic memory”, M.S. Pierce et al, Phys. Rev. Lett. 94, 017202 (2005).
  3. “Return to return point memory”, J.M. Deutsch, A. Dhar and O. Narayan, Phys. Rev. Lett. 92, 227203 (2004).
  4. “Subharmonics and aperiodicity in hysteresis loops”, J.M. Deutsch and O. Narayan, Phys. Rev. Lett. 91, 200601 (2003).

Granular materials

The behavior of granular matter is unexpected in many ways. As a stationary heap, a granular medium supports an externally applied load in a highly inhomogeneous manner: the load is supported by chains of particles that form a ‘scaffolding’, leaving large regions between them that are completely unaffected by the load. As a flowing stream, the collisions between the individual grains are strongly inelastic, leading to inhomogeneous flow and sometimes the flow stopping due to jamming. We have constructed a simple model for how stresses are supported in a stationary granular heap. This model considered a simplified description where interparticle forces were scalars. A vector model can be successfully constructed, which casts doubt on the validity of the Mohr Coulomb failure criterion for poured granular heaps. This criterion is an important ingredient in engineering analyses of the stress distributions in granular media.

  1. “Vector lattice model for stresses in granular materials”, O. Narayan, Phys. Rev. E 63, 010301 (2001).
  2. “Incipient failure in sandpile models”, O. Narayan and S.R. Nagel, Physica A 264, 75 (1999).
  3. “Model for force fluctuations in bead packs”, S.N. Coppersmith, C-h. Liu, S. Majumdar, O. Narayan and T.A. Witten, Phys. Rev. E 53, 4673 (1996).
  4. “Force fluctuations in bead packs”, C-h. Liu et al, Science 269, 513 (1995).

Spin chains

For one-dimensional quantum spin chains, correlation functions can often be computed exactly. For the supersymmetric t-J model, we have obtained a closed form analytical expression for the density matrix. For a class of spin chain wavefunctions with spin s that naturally generalizes the wavefunction for a spin 1/2 chain with 1/r^2 interactions, we have verified numerically that the correlation functions decay algebraically, with a decay exponent 3/(2 s + 2). This is accomplished without any parameters being tuned.

  1. “Exact density matrix of the Gutzwiller wave function as the ground state of the inverse-square supersymmetric t-J model. II. Minority spin component”, O. Narayan, Y. Kuramoto and M. Arikawa, Phys. Rev. B 77, 045114 (2008).
  2. “Exact density matrix of the Gutzwiller wave function as the ground state of the inverse-square supersymmetric t-J model”, O. Narayan and Y. Kuramoto, Phys. Rev. B 73, 195116 (2006).
  3. “Spin-s wavefunctions with algebraic order”, O. Narayan and B.S. Shastry, Phys. Rev. B 70, 184440 (2004).
  4. “The 2-d Coulomb gas on a 1-d lattice”, O. Narayan and B.S. Shastry, J. Phys. A 32, 1131 (1999).

Basic quantum and statistical mechanics

  1. “Influence of a mesoscopic bath on quantum coherence”, O. Narayan and H. Mathur, Phys. Rev. B 72, 045338 (2005).
  2. “Reexamination of experimental tests of the fluctuation theorem”, O. Narayan and A. Dhar, J. Phys. A 37, 63 (2004).
  3. “Convergence of Monte Carlo simulations to equilibrium”, O. Narayan and A.P. Young, Phys. Rev. E 64, 021104 (2001).
  4. “Spin precession and energy conservation”, O. Narayan, Phys. Rev. A 62, 042101 (2000).
  5. “Dyson’s Brownian motion and universal dynamics of quantum systems”, O. Narayan and B.S. Shastry, Phys. Rev. Lett. 71, 2106 (1993).

Depinning transitions

In many systems, the transport when an external driving force is applied can be described as the collective motion of an effectively elastic medium that is locally pinned by impurities. Examples include the motion of vortices in dirty type-II superconductors, the flow of current in weakly disordered charge density waves, and the invasion of a fluid under pressure into a porous medium. We performed an analytical treatement of the ‘depinning transition’ in these systems that separates the low-driving pinned phase from the high-driving moving phase. This was done by using a functional renormalization group (RG) that kept track of an infinite number of coupling constants. In contrast to usual RG studies, short timescale dynamics play a crucial role in determining the scaling exponents of long wavelength low frequency phenomena. We also constructed an analytical description of the distribution of avalanches in the pinned phase, where the external force causes local rearrangements. Connections were established to self-organized criticality, and the apparent different scaling of some quantities explained. Interestingly, the behavior in the strongly disordered limit of these systems is qualitatively different. We developed a new model to treat this strongly disordered regime, establishing connections with percolation theory and predicting the existence of large scale spatial structures.

  1. “Anomalous scaling in depinning transitions”, O. Narayan, Phys. Rev. E 62, R7563 (2000).
  2. “Self similar Barkhausen noise in magnetic domain wall motion”, O. Narayan, Phys. Rev. Lett. 77, 3855 (1996).
  3. “Nonlinear fluid flow in random media: critical phenomena near threshold”, O. Narayan and D.S. Fisher, Phys. Rev. B 49, 9469 (1994).
  4. “Avalanches and the renormalization group for pinned charge-density waves”, O. Narayan and A.A. Middleton, Phys. Rev. B 49, 244 (1994).
  5. “Threshold critical dynamics of driven interfaces in random media”, O. Narayan and D.S. Fisher, Phys. Rev. B 48, 7030 (1993).
  6. “Critical behavior of sliding charge-density waves in 4-e dimensions”, O. Narayan and D.S. Fisher, Phys. Rev. B 46, 11520 (1992).
  7. “Dynamics of sliding charge-density waves in 4-e dimensions”, O. Narayan and D.S. Fisher, Phys. Rev. Lett. 68, 3615 (1992).