Rakesh P. Tiwari
ContactDepartment of Physics
University of Basel
CH-4056 Basel, Switzerland
|2012 - present||Postdoc in the group of Prof. C. Bruder at, University of Basel, Switzerland|
|2010 - 2011||Postdoc in the group of Prof. M. Blaauboer, Delft University of Technology, The Netherlands|
|2004 - 2010||PhD in Physics under the supervision of Prof. David G. Stroud, The Ohio State University, United States|
|2000 - 2004||Bachelor of Technology in Engineering Physics, Indian Institute of Technology, Mumbai, India. Bachelor Thesis Advisor: Prof. Alok Shukla|
- Quantum transport in low dimensions
- Condensed matter realizations of Majorana fermions, parafermions
- Topological insulators, graphene
- Cooper pair splitters
- Superconducting qubits, Quantum computing and Quantum information
- Magnonic, electronic and photonic crystals
- Nanomechanical oscillators
Fall Semester 2016: Nanophysics (VV 11016)
|Lecture 1 (30/11/2016, 8:15-10:00, Auditorium 2, Pharmazentrum)|
|Lecture 2 (7/12/2016, 8:15-10:00, Auditorium 2, Pharmazentrum)|
|Exercise 1 (9/12/2015, 12:00-13:00 AM, 3.12 Physik)|
|Lecture 3 (14/12/2016, 8:15-10:00 AM, Auditorium 2, Pharmazentrum)|
|Exercise 2 (16/12/2015, 12:00-13:00 AM, 3.12 Physik)|
PublicationsShow all abstracts.
|1.||Robust quantum optimizer with full connectivity|
|Simon E. Nigg, Niels Loerch, and Rakesh P. Tiwari.|
Quantum phenomena have the potential to speed up the solution of hard optimization problems. For example quantum annealing, based on the quantum tunneling effect, has recently been shown to scale exponentially better with system size as compared with classical simulated annealing. However, current realizations of quantum annealers with superconducting qubits face two major challenges. First, the connectivity between the qubits is limited, excluding many optimization problems from a direct implementation. Second, decoherence degrades the success probability of the optimization. We address both of these shortcomings and propose an architecture in which the qubits are robustly encoded in continuous variable degrees of freedom. Remarkably, by leveraging the phenomenon of flux quantization, all-to-all connectivity is obtained without overhead. Furthermore, we demonstrate the robustness of this architecture by simulating the optimal solution of a small instance of the NP-hard and fully connected number partitioning problem in the presence of dissipation.
|2.||Dynamic response functions and helical gaps in interacting Rashba nanowires with and without magnetic fields|
|Christopher Pedder, Tobias Meng, Rakesh P. Tiwari, and Thomas L. Schmidt.|
A partially gapped spectrum due to the application of a magnetic field is one of the main probes of Rashba spin-orbit coupling in nanowires. Such a "helical gap" manifests itself in the linear conductance, as well as in dynamic response functions such as the spectral function, the structure factor, or the tunnelling density of states. In this paper, we investigate theoretically the signature of the helical gap in these observables with a particular focus on the interplay between Rashba spin-orbit coupling and electron-electron interactions. We show that in a quasi-one-dimensional wire, interactions can open a helical gap even without magnetic field. We calculate the dynamic response functions using bosonization, a renormalization group analysis, and the exact form factors of the emerging sine-Gordon model. For special interaction strengths, we verify our results by refermionization. We show how the two types of helical gaps, caused by magnetic fields or interactions, can be distinguished in experiments.
|3.||Intrinsic Anomalous Hall Effect in Type-II Weyl Semimetals|
|A. A. Zyuzin and Rakesh P. Tiwari.|
JETP Lett. 103, 717 (2016)
Recently, a new type of Weyl semimetal called type-II Weyl semimetal has been proposed. Unlike the usual (type-I) Weyl semimetal, which has a point-like Fermi surface, this new type of Weyl semimetal has a tilted conical spectrum around the Weyl point. Here we calculate the anomalous Hall conductivity of a Weyl semimetal with a tilted conical spectrum for a pair of Weyl points, using the Kubo formula. We find that the Hall conductivity is not universal and can change sign as a function of the parameters quantifying the tilts. Our results suggest that even for the case where the separation between the Weyl points vanishes, tilting of the conical spectrum could give rise to a finite anomalous Hall effect, if the tilts of the two cones are not identical.
|4.||Non local quantum state engineering with the Cooper pair splitter beyond the Coulomb blockade regime|
|Ehud Amitai, Rakesh P. Tiwari, Stefan Walter, Thomas L. Schmidt, and Simon E. Nigg.|
Phys. Rev. B 93, 075421 (2016)
A Cooper pair splitter consists of two quantum dots side-coupled to a conventional superconductor. Usually, the quantum dots are assumed to have a large charging energy compared to the superconducting gap, in order to suppress processes other than the coherent splitting of Cooper pairs. In this work, in contrast, we investigate the limit in which the charging energy is smaller than the superconducting gap. This allows us, in particular, to study the effect of a Zeeman field comparable to the charging energy. We find analytically that in this parameter regime the superconductor mediates an inter-dot tunneling term with a spin symmetry determined by the Zeeman field. Together with electrostatically tunable quantum dots, we show that this makes it possible to engineer a spin triplet state shared between the quantum dots. Compared to previous works, we thus extend the capabilities of the Cooper pair splitter to create entangled non local electron pairs.
|5.||Snake states and their symmetries in graphene|
|Yang Liu, Rakesh P. Tiwari, Matej Brada, C. Bruder, F.V. Kusmartsev, and E.J. Mele.|
Phys. Rev. B 92, 235438 (2015)
Snake states are open trajectories for charged particles propagating in two dimensions under the influence of a spatially varying perpendicular magnetic field. In the quantum limit they are protected edge modes that separate topologically inequivalent ground states and can also occur when the particle density rather than the field is made nonuniform. We examine the correspondence of snake trajectories in single-layer graphene in the quantum limit for two families of domain walls: (a) a uniform doped carrier density in an antisymmetric field profile and (b) antisymmetric carrier distribution in a uniform field. These families support different internal symmetries but the same pattern of boundary and interface currents. We demonstrate that these physically different situations are gauge equivalent when rewritten in a Nambu doubled formulation of the two limiting problems. Using gauge transformations in particle-hole space to connect these problems, we map the protected interfacial modes to the Bogoliubov quasiparticles of an interfacial one-dimensional p-wave paired state. A variational model is introduced to interpret the interfacial solutions of both domain wall problems.
|6.||8$\pi$ - periodic Josephson effect in time-reversal invariant interacting Rashba nanowires|
|Chris J. Pedder, Tobias Meng, Rakesh P. Tiwari, and Thomas L. Schmidt.|
We investigate narrow quantum wires with strong Rashba spin-orbit coupling and electron-electron interactions. We show that virtual transitions between subbands lead to umklapp scattering which can open a partial gap in the spectrum even in the presence of time-reversal symmetry. Using the superconducting proximity effect to gap out the remaining modes, we show that the system can host zero-energy states at its edges, which are protected by time-reversal symmetry. We present the parameter regime in which these bound states will emerge. Similarly to Majorana bound states, they will produce a zero-bias peak in the differential conductance. In contrast to the Majorana fermions, however, their fourfold degeneracy leads to an 8$\pi$ periodicity of the Josephson current due to tunneling of fractionalized excitations with charge e/2.
|7.||Josephson response of a conventional and a noncentrosymmetric superconductor coupled via a double quantum dot|
|Bjorn Sothmann and Rakesh P Tiwari.|
Phys. Rev. B 92, 014504 (2015)
We consider transport through a Josephson junction consisting of a conventional s-wave superconductor coupled via a double quantum dot to a noncentrosymmetric superconductor with both, singlet and triplet pairing. We calculate the Andreev bound state energies and the associated Josephson current. We demonstrate that the current-phase relation is a sensitive probe of the singlet-triplet ratio in the noncentrosymmetric superconductor. In particular, in the presence of an inhomogeneous magnetic field the system exhibits a $\phi$-junction behavior.
|8.||Detecting nonlocal Cooper pair entanglement by optical Bell inequality violation|
|Simon E. Nigg, Rakesh P. Tiwari, Stefan Walter, and Thomas L. Schmidt.|
Phys. Rev. B 91, 094516 (2015)
Based on the Bardeen Cooper Schrieffer (BCS) theory of superconductivity, the coherent splitting of Cooper pairs from a superconductor to two spatially separated quantum dots has been predicted to generate nonlocal pairs of entangled electrons. In order to test this hypothesis, we propose a scheme to transfer the spin state of a split Cooper pair onto the polarization state of a pair of optical photons. We show that the produced photon pairs can be used to violate a Bell inequality, unambiguously demonstrating the entanglement of the split Cooper pairs.
|9.||Non-Abelian parafermions in time-reversal invariant interacting helical systems|
|Christoph P. Orth, Rakesh P Tiwari, Tobias Meng, and Thomas L. Schmidt.|
Phys. Rev. B 91, 081406 (2015)
The interplay between bulk spin-orbit coupling and electron-electron interactions produces umklapp scattering in the helical edge states of a two-dimensional topological insulator. If the chemical potential is at the Dirac point, umklapp scattering can open a gap in the edge state spectrum even if the system is time-reversal invariant. We determine the zero-energy bound states at the interfaces between a section of a helical liquid which is gapped out by the superconducting proximity effect and a section gapped out by umklapp scattering. We show that these interfaces pin charges which are multiples of e/2, giving rise to a Josephson current with 8\pi periodicity. Moreover, the bound states, which are protected by time-reversal symmetry, are fourfold degenerate and can be described as Z4 parafermions. We determine their braiding statistics and show how braiding can be implemented in topological insulator systems.
|10.||Josephson effect in normal and ferromagnetic topological insulator planar, step and edge junctions|
|Jennifer Nussbaum, Thomas L. Schmidt, Christoph Bruder, and Rakesh P. Tiwari.|
Phys. Rev. B 90, 045413 (2014)
We investigate Josephson junctions on the surface of a three-dimensional topological insulator in planar, step, and edge geometries. The elliptical nature of the Dirac cone representing the side surface states of the topological insulator results in a scaling factor in the Josephson current in a step junction as compared to the planar junction. In edge junctions, the contribution of the Andreev bound states to the Josephson current vanishes due to spin-momentum locking of the surface states. Furthermore, we consider a junction with a ferromagnetic insulator between the superconducting regions. In these ferromagnetic junctions, we find an anomalous finite Josephson current at zero phase difference if the magnetization is pointing along the junction (and perpendicular to the Josephson current). An out-of-plane magnetization with respect to the central region of the junction opens up an exchange gap and leads to a non-monotonic behavior of the critical Josephson current for sufficiently large magnetization as the chemical potential increases.
|11.||Quantum charge pumping through fractional Fermions in charge density modulated quantum wires and Rashba nanowires|
|Arijit Saha, Diego Rainis, Rakesh P Tiwari, and Daniel Loss.|
Phys. Rev. B 90, 035422 (2014)
We study the phenomenon of adiabatic quantum charge pumping in systems supporting fractionally charged fermionic bound states, in two different setups. The first quantum pump setup consists of a charge-density-modulated quantum wire, and the second one is based on a semiconducting nanowire with Rashba spin-orbit interaction, in the presence of a spatially oscillating magnetic field. In both these quantum pumps transport is investigated in a N-X-N geometry, with the system of interest (X) connected to two normal-metal leads (N), and the two pumping parameters are the strengths of the effective wire-lead barriers. Pumped charge is calculated within the scattering matrix formalism. We show that quantum pumping in both setups provides a unique signature of the presence of the fractional-fermion bound states, in terms of asymptotically quantized pumped charge. Furthermore, we investigate shot noise arising due to quantum pumping, verifying that quantized pumped charge corresponds to minimal shot noise.
|12.||Neutral edge modes in a superconductor -- topological-insulator hybrid structure in a perpendicular magnetic field|
|Rakesh P Tiwari, U Zulicke, C Bruder, and Vladimir M. Stojanovic.|
EPL 108, 17009 (2014).
We study the low-energy edge states of a superconductor -- 3D topological-insulator hybrid structure (NS junction) in the presence of a perpendicular magnetic field. The hybridization of electron-like and hole-like Landau levels due to Andreev reflection gives rise to chiral edge states within each Landau level. We show that by changing the chemical potential of the superconductor, this junction can be placed in a regime where the sign of the effective charge of the edge state within the zeroth Landau level changes more than once resulting in neutral edge modes with a finite value of the guiding-center coordinate. We find that the appearance of these neutral edge modes is related to the level repulsion between the zeroth and the first Landau levels in the spectra. We also find that these neutral edge modes come in pairs, one in the zeroth Landau level and its corresponding pair in the first.
|13.||Signatures of tunable Majorana-fermion edge states|
|Rakesh P Tiwari, U. Zulicke, and C. Bruder.|
New J. Phys. 16, 025004 (2014)
Chiral Majorana-fermion modes are shown to emerge as edge excitations in a superconductor–topological-insulator hybrid structure that is subject to a magnetic field. The velocity of this mode is tunable by changing the magnetic-field magnitude and/or the superconductor's chemical potential. We discuss how quantum-transport measurements can yield experimental signatures of these modes. A normal lead coupled to the Majorana-fermion edge state through electron tunneling induces resonant Andreev reflections from the lead to the grounded superconductor, resulting in a distinctive pattern of differential-conductance peaks.
|14.||Quantum transport signatures of chiral edge states in Sr2RuO4|
|Rakesh P Tiwari, W. Belzig, Manfred Sigrist, and C. Bruder.|
Phys. Rev. B 89, 184512 (2014)
We investigate transport properties of a double quantum dot based Cooper pair splitter, where the superconducting lead consists of Sr$_2$RuO$_4$. The proposed device can be used to explore the symmetry of the superconducting order parameter in Sr$_2$RuO$_4$ by testing the presence of gapless chiral edge states, which are predicted to exist if the bulk superconductor is described by a chiral $p$--wave state. The odd orbital symmetry of the bulk order parameter ensures that we can realize a regime where the electrons tunneling into the double dot system come from the chiral edge states and thereby leave their signature in the conductance. The proposed Cooper pair splitter has the potential to probe order parameters in unconventional superconductors.
|15.||Adiabatic quantum pumping of chiral Majorana fermions|
|M. Alos-Palop, Rakesh P. Tiwari, and M. Blaauboer.|
Phys. Rev. B 89, 045307 (2014).
We investigate adiabatic quantum pumping of chiral Majorana states in a system composed of two Mach--Zehnder type interferometers coupled via a quantum point contact. The pumped current is generated by periodic modulation of the phases accumulated by traveling around each interferometer. Using scattering matrix formalism we show that the pumped current reveals a definite signature of the chiral nature of the Majorana states involved in transport in this geometry. Furthermore, by tuning the coupling between the two interferometers the pump can operate in a regime where finite pumped current and zero two-terminal conductance is expected.
|16.||Majorana fermions from Landau quantization in a superconductor--topological-insulator hybrid structure|
|Rakesh P. Tiwari, U. Zulicke, and C. Bruder.|
Phys. Rev. Lett. 110, 186805 (2013)
We show that the interplay of cyclotron motion and Andreev reflection experienced by massless-Dirac-like charge carriers in topological-insulator surface states generates a Majorana-particle excitation. Based on an envelope-function description of the Dirac-Andreev edge states, we discuss the kinematic properties of the Majorana mode and find them to be possible to be tuned by changing the superconductor's chemical potential and/or the magnitude of the perpendicular magnetic field. Our proposal opens up new possibilities for studying Majorana fermions in a controllable setup.
|17.||Suppression of Conductance in a Topological Insulator Nanostep Junction|
|M. Alos-Palop, Rakesh P Tiwari, and M. Blaauboer.|
Phys. Rev. B 87, 035432 (2013)
We investigate quantum transport via surface states in a nanostep junction on the surface of a 3D topological insulator that involves two different side surfaces. We calculate the conductance across the junction within the scattering matrix formalism and find that as the bias voltage is increased, the conductance of the nanostep junction is suppressed by a universal factor of 1/3 compared to the conductance of a similar planar junction based on a single surface of a topological insulator. We also calculate and analyze the Fano factor of the nanostep junction and predict that the Fano factor saturates at 1/5, five times smaller than for a Poisson process.
|18.||Adiabatic quantum pumping through surface states in 3D topological insulators|
|M. Alos-Palop, Rakesh P. Tiwari, and M. Blaauboer.|
New J. Phys. 14, 113003 (2012)
We investigate adiabatic quantum pumping of Dirac fermions on the surface of a strong 3D topological insulator. Two different geometries are studied in detail, a normal metal -- ferromagnetic -- normal metal (NFN) junction and a ferromagnetic -- normal metal -- ferromagnetic (FNF) junction. Using a scattering matrix approach, we first calculate the tunneling conductance and then the adiabatically pumped current using different pumping mechanisms for both types of junctions. We explain the oscillatory behavior of the conductance by studying the condition for resonant transmission in the junctions and find that each time a new resonant mode appears in the transport window, the pumped current diverges. We also predict an experimentally distinguishable difference between the pumped current and the rectified current.
|19.||Localization and circulating currents in curved graphene devices|
|G. M. M. Wakker, Rakesh P Tiwari, and M. Blaauboer.|
Phys. Rev. B 84, 195427 (2011).
We calculate the energy spectrum and eigenstates of a graphene sheet that contains a circular deformation. Using time-independent perturbation theory with the ratio of the height and width of the deformation as the small parameter, we find that due to the curvature the wave functions for the various states acquire unique angular asymmetry. We demonstrate that the pseudomagnetic fields induced by the curvature result in circulating probability currents.
|20.||Quantum pumping in graphene with a perpendicular magnetic field|
|Rakesh P Tiwari and M. Blaauboer.|
Appl. Phys. Lett. 97, 243112 (2010).
We consider quantum pumping of Dirac fermions in a monolayer of graphene in the presence of a perpendicular magnetic field in the central pumping region. The two external pump parameters are electrical voltages applied to the graphene sheet on either side of the pumping region. We analyze this pump within scattering matrix formalism and calculate both pumped charge and spin currents. The predicted charge currents are of the order of 1000 nA, which is readily observable using current technology.
|21.||Magnetic superlattice with two-dimensional periodicity as a waveguide for spin waves|
|Rakesh P Tiwari and D. Stroud.|
Phys. Rev. B 81, 220403(R) (2010).
We describe a simple method of including dissipation in the spin-wave band structure of a periodic ferromagnetic composite, by solving the Landau-Lifshitz equation for the magnetization with the Gilbert damping term. We use this approach to calculate the band structure of square and triangular arrays of Ni nanocylinders embedded in an Fe host. The results show that there are certain bands and special directions in the Brillouin zone where the spin-wave lifetime is increased by more than an order of magnitude above its average value. Thus, it may be possible to generate spin waves in such composites which decay especially slowly, and propagate especially large distances, for certain frequencies and directions in k space.
|22.||Tunable band gap in graphene with a noncentrosymmetric superlattice potential|
|Rakesh P. Tiwari and D. Stroud.|
Phys. Rev. B 79, 205435 (2009).
We show that, if graphene is subjected to the potential from an external superlattice, a band gap develops at the Dirac point provided the superlattice potential has broken inversion symmetry. As numerical example, we calculate the band structure of graphene in the presence of an external potential due to periodically patterned gates arranged in a triangular graphene superlattice (TGS) or a square graphene superlattice with broken inversion symmetry, and find that a band gap is created at the original and, in the case of a TGS, the “second generation” Dirac point. This gap, which extends throughout the superlattice Brillouin zone, can be controlled, in principle, by changing the external potential and the lattice constant of the superlattice. For a square superlattice of lattice-constant 10 nm, we have obtained a gap as large as 65 meV, for gate voltages no larger than 1.5 V.
|23.||Model for the magnetoresistance and Hall coefficient of inhomogeneous graphene|
|Rakesh P Tiwari and D. Stroud.|
Phys. Rev. B 79, 165408 (2009).
We show that when bulk graphene breaks into n-type and p-type puddles, the in-plane resistivity becomes strongly field dependent in the presence of a perpendicular magnetic field even if homogeneous graphene has a field-independent resistivity. We calculate the longitudinal resistivity ρxx and Hall resistivity ρxy as a function of field for this system using the effective-medium approximation. The conductivity tensors of the individual puddles are calculated using a Boltzmann approach suitable for the band structure of graphene near the Dirac points. The resulting resistivity agrees well with experiment provided that the relaxation time is weakly field dependent. The calculated Hall resistivity has the sign of the carriers in the puddles occupying the greater area of the composite and vanishes when there are equal areas of n- and p-type puddles.
|24.||Sound propagation in light-modulated carbon nanosponge suspensions|
|W. Zhou, Rakesh P Tiwari, R. Annamalai, R. Sooryakumar, V. Subramaniam, and D. Stroud.|
Phys. Rev. B 79, 104204 (2009).
Single-walled carbon nanotube bundles dispersed in a highly polar fluid are found to agglomerate into a porous structure when exposed to low levels of laser radiation. The phototunable nanoscale porous structures provide an unusual way to control the acoustic properties of the suspension. Despite the high sound speed of the nanotubes, the measured speed of longitudinal-acoustic waves in the suspension decreases sharply with increasing bundle concentration. Two possible explanations for this reduction in sound speed are considered. One is simply that the sound speed decreases because of fluid heat induced by laser light absorption by the carbon nanotubes. The second is that this decrease results from the smaller sound velocity of fluid confined in a porous medium. Using a simplified description of convective heat transport, we estimate that the increase in temperature is too small to account for the observed decrease in sound velocity. To test the second possible explanation, we calculate the sound velocity in a porous medium, using a self-consistent effective-medium approximation. The results of this calculation agree qualitatively with experiment. In this case, the observed sound wave would be the analog of the slow compressional mode of porous solids at a structural length scale of order of 100 nm.
|25.||Numerical study of energy loss by a nanomechanical oscillator coupled to a Cooper-pair box|
|Rakesh P Tiwari and D. Stroud.|
Phys. Rev. B 77, 214520 (2008).
We calculate the dynamics of a nanomechanical oscillator (NMO) coupled capacitively to a Cooper-pair box (CPB) by solving a stochastic Schrödinger equation with two Lindblad operators [ Commun. Math. Phys. 48 119 (1976)]. Both the NMO and the CPB are assumed dissipative, and the coupling is treated within the rotating wave approximation. We show numerically that, if the CPB decay time is smaller than the NMO decay time, the coupled NMO will lose energy faster and the coupled CPB more slowly than the uncoupled NMO and CPB do. The results show that the efficiency of energy loss by an NMO can be substantially increased if the NMO is coupled to a CPB.
|26.||Suppression of tunneling in a superconducting persistent-current qubit|
|Rakesh P Tiwari and D. Stroud.|
Phys. Rev. B 76, 220505(R) (2007).
We consider a superconducting persistent-current qubit consisting of a three-junction superconducting loop in an applied magnetic field. We show that by choosing the field, Josephson couplings, and offset charges suitably, we can perfectly suppress the tunneling between the two oppositely directed states of circulating current, leading to a vanishing of the splitting between the two qubit states. This suppression arises from interference between tunneling along different paths and is analogous to that predicted previously for magnetic particles with half-integer spin.
|27.||A basis-set based Fortran program to solve the Gross-Pitaevskii Equation for dilute Bose gases in harmonic and anharmonic traps.|
|Rakesh P Tiwari and Alok Shukla.|
Comput. Phys. Commun. 174, 966 (2006).
Inhomogeneous boson systems, such as the dilute gases of integral spin atoms in low-temperature magnetic traps, are believed to be well described by the Gross–Pitaevskii equation (GPE). GPE is a nonlinear Schrodinger equation which describes the order parameter of such systems at the mean field level. In the present work, we describe a Fortran 90 computer program developed by us, which solves the GPE using a basis set expansion technique. In this technique, the condensate wave function (order parameter) is expanded in terms of the solutions of the simple-harmonic oscillator (SHO) characterizing the atomic trap. Additionally, the same approach is also used to solve the problems in which the trap is weakly anharmonic, and the anharmonic potential can be expressed as a polynomial in the position operators x, y, and z. The resulting eigenvalue problem is solved iteratively using either the self-consistent-field (SCF) approach, or the imaginary time steepest-descent (SD) approach. Iterations can be initiated using either the simple-harmonic-oscillator ground state solution, or the Thomas–Fermi (TF) solution. It is found that for condensates containing up to a few hundred atoms, both approaches lead to rapid convergence. However, in the strong interaction limit of condensates containing thousands of atoms, it is the SD approach coupled with the TF starting orbitals, which leads to quick convergence. Our results for harmonic traps are also compared with those published by other authors using different numerical approaches, and excellent agreement is obtained. GPE is also solved for a few anharmonic potentials, and the influence of anharmonicity on the condensate is discussed. Additionally, the notion of Shannon entropy for the condensate wave function is defined and studied as a function of the number of particles in the trap. It is demonstrated numerically that the entropy increases with the particle number in a monotonic way.