by the simple process 
 of joining the ends of a linear polymer chain , 
 one obtains a ring polymer ( rp ) . while 
 the architecture of ring polymers is very simple , they differ in many 
 interesting ways from their linear counterparts and are the subject 
 of active research in physics , biology , chemistry , and even pure mathematics . one interesting consequence is that for a dynamics that disallows 
 strand crossing , there are different classes of configurations of 
 a rp , which can never transform into each other . these are referred 
 to as the topology classes or knot types of a rp . knot theory is a fascinating and active branch of mathematics 
 with many open problems concerning the enumeration and classification 
 of knots . a rp that has the topology of 
 a circle is called an unknotted rp . unlike 
 rps , linear polymer chains can strictly speaking never be 
 knotted , as every configuration of a linear polymer chain can be continuously 
 transformed to a straight line , without the need for strand crossings . while this is a fundamental difference between linear polymer chains 
 and rps , it is nevertheless possible to extend the concept of knots 
 to physical knots on linear polymer chains . there are many works dealing with the properties 
 of these physical knots on linear polymer chains , in particular due to their relevance in biophysics , where they are 
 for instance found in dna and can have significant 
 effects on key processes . the topological constraint 
 of a ring polymer has important consequences 
 for its physical behavior . it took the work of many authors to establish that the diameter of gyration dg of an isolated ideal rp with fixed topology scales as dg2 n , where  
  0.588 is the flory exponent , which also describes the scaling 
 behavior of the radius of gyration of self - avoiding linear chains . this is true for all knot types , even for an 
 ideal unknotted rp ( i.e. , without monomer excluded volume interactions , 
 just keeping the topological constraints ) . an ideal linear polymer 
 chain , on the other hand , remains more compact and exhibits the scaling 
 law of a nonself - avoiding random walk dg2 n. another important difference lies in the effective potential 
 between the centers of mass of rps . while the effective potential 
 vanishes between infinitely thin linear polymer chains and also between 
 an infinitely thin linear polymer chain and a rp , there remains a 
 nonzero repulsive contribution between cyclic polymers with fixed 
 topology . furthermore , it was shown that the effective potential 
 between two moderately sized rps increases with the knot complexity 
 of the rings . also for concentrated 
 systems , the topology of polymer chains plays 
 an important role . the scaling of linear polymer chains in the melt 
 is the one of a nonself - avoiding random walk dg2 n. simulations of dense systems of rps , on the other hand , showed that while short chains also exhibit a 
 gaussian scaling behavior dg2  n , long chains are compact and thus scale as dg2  n. in between there 
 is a broad crossover region , where a dg2  n scaling provides a good description of the 
 data . for the dynamics , it is expected that concatenations of ring 
 polymers can have a significant effect , as they are permanent , in 
 contrast to the entanglement of linear polymer chains . however , this 
 implies that those concatenations are there in the first place , i.e. , 
 from the very synthesis of the sample on . even in the absence of concatenations , 
 there are important differences in the dynamics of rps in the melt 
 with respect to their linear counterparts . for instance , recent experiments and simulations revealed a power - law stress relaxation instead of the rubbery plateau 
 found for linear chains . for the large intermediate density 
 domain between dilute solutions 
 and melts , there are relatively few theoretical results despite the 
 practical relevance of this regime for instance in the field of biophysics , 
 where the topological interactions between chromatin loops plays a 
 crucial role in the creation of chromosome territories . a fruitful and modern approach for the economic description 
 and simulation of macromolecules in this regime is the method of coarse - graining . the idea behind this method is to bridge the time and length scales 
 in the system by describing the macromolecules via an effective model 
 with a reduced set of suitably chosen effective degrees of freedom 
 ( dof ) . the microscopic information on the monomer - resolved model is 
 underlying the effective model , as it determines the form of the effective 
 potential , which describes the interaction between the macromolecules . the advantage of this method is not only that every time step in a 
 simulation requires less computational effort due to the simplified 
 representation but also that one can often choose a much larger time 
 step in a simulation of the coarse grained model , as the dof that 
 remain in the coarse - grained model change much slower in time than 
 their counterparts in the monomer - resolved model . the method of coarse - graining is well - established and has 
 for instance 
 found successful application for polymer chains , star polymers , star - shaped polyelectrolytes , dendrimers , and block copolymers . the identification of the relevant 
 degrees of freedom is an essential part in the design of an effective 
 model . one often uses isotropic effective models , where the macromolecules 
 consisting of many individual monomers are reduced to their center 
 of mass . for the semiflexible ring polymers such a model clustering was observed in monomer - resolved simulations 
 of semiflexible ring polymers as well as in the corresponding isotropic 
 effective model . however , it was also shown that the monomer - resolved 
 system shows anisotropic features that can not be accounted for in 
 the isotropic effective model . also , the correlation functions stemming 
 from the isotropic effective model are markedly different from the 
 microscopically derived ones . anisotropy is particularly strong for 
 rings with high bending stiffness or few monomers , as they have a 
 strong tendency to orient with respect to other rings in their proximity . this motivates us to introduce an anisotropic effective model for 
 the description of semiflexible ring polymers in this article . in 
 this model , we will define the effective particles as soft disk - like 
 molecules which are described not only by their center of mass but 
 also by the direction in which their faces are oriented . an anisotropic 
 effective model was already used successfully for the description 
 of hard disk - like macromolecules , but 
 to the best of our knowledge this approach was up to now never applied 
 to penetrable macromolecules , where the centers of mass of the macroparticles 
 can coincide . penetrable particles are particularly interesting as 
 they allow for clustering , which often leads to a rich phase behavior . 
 for instance , point - particles interacting with a certain class of 
 ultrasoft potentials form so - called cluster crystals . unlike in an ordinary crystal multiple soft particles can sit on 
 top of each other at the same lattice site in a cluster crystal . another 
 peculiar feature of this state of matter is that by compressing it 
 one only changes the occupation number of particles per lattice site , 
 while the lattice constant remains invariant . monomer - resolved simulations 
 of semiflexible ring polymers , on the other hand , show the formation 
 of the cluster glass phase , which is an arrested 
 state that also contains some of the features found in the cluster 
 crystal phase . in both cases the overall structure of the system is 
 frozen , while individual particles can hop between the lattice sites 
 of the cluster crystal or the stacks found in the cluster glass . elongated 
 dendrimers , which unlike hard rod - like particles exhibit local antinematic 
 order , are another interesting example 
 for a system of penetrable particles , which behaves distinctively 
 different to its solid counterpart . by creating an anisotropic effective 
 model for the semiflexible ring polymers , we aim at a model that is 
 still computationally cheap and that improves the description obtained 
 by the isotropic model , especially for the case of high densities . in addition , the analysis of the interactions in the anisotropic effective 
 potential allows us to get a better understanding for the interaction 
 between the anisotropic , penetrable nanoparticles in systems of semiflexible 
 rps . we first 
 present 
 the hamiltonian of the monomer - resolved model which we use for the 
 description of semiflexible rps and then introduce an anisotropic 
 effective model for such a system . in section a of the appendix we give more details about the derivation of the 
 effective interactions for such a model . we carried out molecular 
 dynamics simulations of the monomer - resolved model and monte carlo 
 simulations of the effective models ; details concerning these simulations 
 are given in section 3 . in section 4 we 
 present the anisotropic effective potential and discuss its features . results of monte carlo simulations with this potential , which show 
 that the inclusion of anisotropy in the effective model can significantly 
 improve the agreement with the monomer - resolved model , are presented 
 in section 5 , whereas 
 in section 6 we briefly 
 discuss the effects of truncation of the expansion of the potential 
 on the quality of the results . in the appendix , we explain the expansion of the anisotropic pair - correlation 
 function of a system of two rps , which contains all the information 
 for calculating the effective potential , as a sum of suitably chosen 
 basis functions . the derivation of the anisotropic effective model 
 is based on a microscopic model of semiflexible ring polymers , each 
 consisting of n monomers . thus , any two monomers 
 interact via the truncated and shifted lennard - jones potential1this potential 
 is purely repulsive , accounting 
 then for monomer excluded volume interactions . bonded monomers also 
 interact through a finitely extensible nonlinear elastic potential 
 ( fene)2rigidity is introduced via the bending potential3where  is the angle between two consecutive 
 bond vectors . [ note that this is not the kratky  porod model 
 ( linear in the cosine ) . we expect the same qualitative results for 
 kratky  porod rings with the same n s 
 and persistence lengths as the model simulated here . ] the potential vbend vanishes for  = 0 , when the polymer 
 chain does not bend at the respective angle . we choose  = kbt , k = 30kbt/ , r0 = 1.5 , and  = 30kbt , where kb is the boltzmann constant and t the temperature . these are precisely the parameters employed in the simulation study 
 of ref ( 52 ) . the corresponding 
 dynamics does not allow for chain crossings and thus topology is preserved . in ref ( 58 ) the 
 characteristic ratio of this polymer 
 model was estimated by carrying out simulations of isolated linear 
 chains . excluded volume interactions were switched off except for 
 mutually connected monomers in order to obtain long - range gaussian 
 statistics . c was obtained by 
 analyzing the long - s limit of the ratio r(s)/sb , where r(s ) is the distance between two monomers i , j with s = |i  j| , and b is 
 the bond length ( b = 
 0.94 ) . the authors reported a value of c  15 , which is typical for stiff polymers . we can give an estimate for the persistence length of the 
 model by mapping it to the freely rotating chain model using the relation 
 cos  = ( c  1)/(c + 1 ) , where  is the bending 
 angle of the freely rotating chain model . the persistence length is then obtained as spb = b / ln(cos  )  7.3 . we carried out simulations of ring 
 polymers with n = 20 , 50 , and 100 monomers , which 
 have the contour to persistence length ratio n / sp  2.7 , 6.7 , and 13.3 , 
 respectively . in earlier 
 work , bernabei et al . carried out extensive 
 monomer - resolved simulations of stiff ring polymers to obtain the 
 structure of concentrated solutions of the same . in an attempt to 
 coarse - grain the system in the simplest possible way , they also derived 
 and employed an isotropic effective potential for their effective 
 description , reducing thereby stiff rp s into point - like effective 
 particles , namely their centers of mass . at this level of approximation , the effective particles possess no other internal ( spin - like ) degrees 
 of freedom , and thus the effective interaction is isotropic . the effective 
 potential between these macroparticles was defined by calculating 
 the pair correlation function g(r ) in an infinitely dilute system and using4to define the effective interaction potential 
 between these point particles , where  = 1/(kbt ) . in the infinitely dilute case , the 
 distribution of the centers of mass of the ring polymers in equilibrium 
 is identical to the distribution of the point particles in the effective 
 model . at higher densities , however , it turns out that multiparticle 
 terms in the effective potential are necessary to obtain the correct 
 equilibrium distribution of the centers of mass in the effective model . one of the reasons due to which multiparticle interactions become 
 important is that two ring polymers that are sufficiently close and 
 stiff will prefer to align parallel to each other . a third ring polymer 
 interacting with those two will not see them as two independent rings 
 but as a system of two rings that are correlated . when using the potential veff(r ) calculated in eq 4 , one assumes that the 
 free energy penalty of one ring with respect to a second is independent 
 of the presence of another polymer in the vicinity of the second . if the ring polymers preferentially align parallel , this assumption 
 is clearly violated and one has to correct the effective potential 
 by introducing multiparticle terms . showed that already 
 at moderate densities one can encounter strong correlations of the 
 orientations of the semiflexible ring polymers , in particular if the 
 chains contain only few monomers ( e.g. , n = 20 ) . therefore , a more accurate coarse - graining which 
 takes into account the ring anisotropy is called for . the easiest 
 way to incorporate the correlations of the orientations of rings is 
 to introduce them via additional degrees of freedom ( dof ) in the effective 
 description . , we need to first come up with a suitable definition for the 
 orientation of a rp . to this end , we make use of the gyration tensor5where r(i ) (  
 = x , y , z , cartesian 
 components ) denotes the position of the ith monomer 
 with respect to the center of mass of the ring to which this monomer 
 belongs . the eigenvectors of s are the principal axes of an ellipsoid that approximates the shape 
 of the macromolecule : if a rp is flat , which means that all its monomers 
 lie in one plane , the ellipsoid has one zero eigenvalue with a corresponding 
 eigenvector that is perpendicular to that plane . also in the more 
 general case , where the monomers do not all lie in the same plane , 
 we define the normalized eigenvector corresponding to the smallest 
 eigenvalue of s as the direction 
 vector d of the rp . the ring polymers 
 in the anisotropic effective model we propose are described via the 
 position vectors of their centers of mass , r , and their direction vector , d. henceforth , we describe the stiff rings 
 as soft circular disks and ignore differences in the other two eigenvectors 
 of the gyration tensor s. this choice is motivated by the limit of infinite bending stiffness , 
 where the rings assume flat and precisely circular conformations . our model therefore amounts to the minimal anisotropic 
 extension of the spherically symmetric effective interaction between 
 the centers of mass . we emphasize , however , that there is no a priori guarantee that this will be an improvement over 
 the isotropic model at finite densities and in particular at high 
 concentrations : this depends on the degree in which the rp s 
 at high concentration maintain their anisotropic shape and properties 
 encoded in the high - dilution limit in which the anisotropic pair potential 
 is derived . accordingly , the introduction of such a potential is not 
 a straightforward part of a systematic strategy of introducing more 
 and more detail into the effective description of the system . in order to determine the anisotropic effective potential veff , we carried out monomer - resolved simulations 
 of two ring polymers inside a large simulation box . the effective 
 pair potential is then defined such that it exactly reproduces the 
 correlation functions of the effective degrees of freedom in this 
 infinitely dilute , monomer - resolved simulation . in the effective model , three 
 for each center of mass and two for each direction vector of each 
 ring polymer . however , due to translation , rotation , and mirror symmetry , 
 the distinct configurations ( those that can not be related by symmetry 
 transformations ) of a system with two effective particles are reduced 
 and can be specified by four parameters only . a convenient choice 
 for these variables is illustrated in figure 1 and reads as follows:6where r  r  r is the connection 
 vector between the centers of mass of the two rings , r  r / r the unit vector 
 in the direction of r , and d(i ) the component of the director d perpendicular to r ; 0   
   denotes the angle between vectors d(1 ) and d(2 ) . by selecting the appropriate sign of d , we can always choose cos i to lie in the interval [ 0 , 1 ] . illustration of the effective 
 variables r = |r| , i , and  with 
 which relative configurations of ring polymers are described . let us define the ideal case as 
 the system where the effective 
 particles do not interact , and thus every orientation and position 
 of the effective particles occur with equal probability , independently 
 of the configuration of the other effective particle . with the effective 
 coordinates defined in eq 6 the probability density in the ideal system , pid(r , cos 1 , cos 2 ,  ) is proportional to r and constant in both cos i and 
 . this simple behavior of pid(r , cos 1 , cos 2 ,  ) , 
 makes eq 6 a particularly 
 convenient choice of the effective coordinates . in the simulations 
 with two ring polymers , we obtain a probability density p(r , cos 1 , cos 2 ,  ) that is different from the ideal distribution pid . we define as a generalized version of the 
 radial distribution function the anisotropic pair correlation function 
 as7thus , 
 the quantity g(r , cos 1 , cos 2 ,  ) 
 describes the factor by which configurations in the effective anisotropic 
 model have to be enhanced or suppressed with respect to the ideal 
 case , in order to obtain a distribution for the effective dof that 
 is identical to the distribution obtained in a monomer - resolved simulation 
 in the infinitely dilute case . as in the isotropic case ( 4 ) the relation to the associated effective potential reads8from the anisotropic effective potential , 
 we can deduce the isotropic pair - correlation function via9the associated isotropic effective potential 
 is then given by ( 4 ) . the effective potential 
 between two identical ring polymers remains 
 invariant if we swap the orientations of their respective director 
 with respect to the connection vector , i.e. , if we swap the values 
 of the polar angles 1 and 2 of 
 the two rings:10this symmetry is violated 
 for the effective 
 potential between bidisperse ring polymers , e.g. , for ring polymers 
 with a different number of monomers . however , apart from this symmetry , 
 there would be no differences in the procedure of calculating the 
 effective potential between different types of ring polymers . to determine the effective potential , we carry 
 out constant nvt molecular dynamics simulations with 
 two ring polymers . 
 we simulate rings of n = 20 , 50 , and 100 monomers . the polymer rings are placed in a simulation 
 box , which is large enough to prevent multiple interactions via the 
 periodic boundary conditions . the corresponding equations 
 of motion read as11here , ri is the position of the ith monomer , m its mass , and fi(t ) is the deterministic 
 force acting on it , which 
 includes the microscopic forces originating from potentials ( 1)(3 ) and the force 
 originating from a biasing potential . the bias potential vbias = ( r  rj)kj/2 introduces a harmonic spring with spring 
 constant kj between the 
 centers of mass of the ring polymers . we 
 carried out simulations for different values of rj , starting at rj = 0 and increasing it up to some maximum 
 value rc in steps of 
 /2 . for n = 20 , 50 , and 100 , these values for rc are much bigger than the infinite - dilution 
 diameters of gyration ( dg0 = 5.9 , 
 13 , and 21.5 for n = 20 , 50 , and 100 , 
 respectively ) . for kj we chose the values 2.5/ and / for all ring sizes , and for n = 20 we also 
 carried out simulations with kj = 5/. the quantity i(t ) is a random force , 
 with i(t) = 0 , which is related to the friction coefficient 
  by the fluctuation dissipation relation i(t)j(t) 
 = 2mkbtij(t  t ) ,  and  
 denoting cartesian components . our unit of time is set by t0 = ( m/ ) , and the friction coefficient  is chosen as 1/t0 . we integrate the equations of motion with 
 a time step of t = 10t0 and use 2  10 timesteps 
 for equilibrating the system and collect data during another 2  
 10 timesteps . we sample histograms p(q ) , where q refers to a bin in the 4d space of the effective coordinates . p(q ) gives 
 the probability for a state in the jth simulation 
 to have effective coordinates in bin q. the histograms 
 have 128 bins in r and 16 bins in cos 1 , cos 2 , and  direction . as discussed 
 in section a of the appendix , we use the self - consistent 
 histogram method by ferrenberg and swendsen to combine the p(q ) histograms for simulations with identical ring 
 sizes but different biasing potentials to arrive at an estimate for p(q ) in the unbiased system . using the 
 anisotropic effective potential , we carry out standard metropolis 
 monte carlo ( mc ) simulations for the anisotropic effective model . the values of the anisotropic effective potential have been calculated 
 on a discrete grid in the ( r , cos 1 , cos 2 ,  ) space , and we use linear interpolation 
 to estimate the values of exp(veff ) in between the grid points . having in mind a comparison 
 with both the monomer - resolved simulation results and the isotropic 
 effective potential of ref ( 52 ) , we choose the same number of particles and effective densities 
 that were used in those simulations . for rings with n = 20 , 50 , and 100 monomers we simulate systems of n = 2400 , 1600 , and 1200 rings , respectively , varying in each case 
 accordingly the cubic box size l as to achieve the 
 desired density  = n / l . as the effective potential veff is bounded , a random distribution of the particles in the simulation 
 box can be used as initial condition . we have implemented two types 
 of mc moves : the first one translates a randomly chosen particle in 
 a random direction , and the second randomly rotates the particle s 
 director by some angle . the distance by which the particles are displaced 
 and the angle by which they are rotated is randomly selected in an 
 interval starting at 0 and going up to some maximum value . for both 
 moves , this maximum value of the interval is chosen such that the 
 acceptance ratio is approximately 15% . we use 9  10 mc moves to equilibrate the system . during this equilibration period the individual soft particles diffuse to several times their own diameter . 
 afterward , during 15  10 mc moves equilibrium configurations 
 are generated . we store the configurations every 20  10 moves and use them to compute the physical observables that 
 are presented in section 5 . the gain in computational efficiency for the simulations 
 in going from a monomer - resolved to a coarse - grained simulation is 
 considerable . this can conveniently be characterized 
 by means of the mean - square displacement of the rings per unit of 
 cpu time . if one ignores the detailed implementation aspects , the 
 cpu time spent on a single sweep over all monomers in the former and 
 a run over all effective ring particles in the latter , which strictly 
 speaking depend on both the number of monomers n per 
 ring and the overall density of rings , are of similar magnitude . however , 
 the diffusion per sweep in the coarse - grained simulation is significantly 
 larger than that for the monomer resolved simulation ; i.e. , for the 
 case of n = 50 and  * = 20 this results in 
 a factor of approximately 10 . first of all , the translation / rotation of 
 an effective ring corresponds to a much more time - consuming collective 
 movement of the constituents . the second even more important contribution 
 arises from the steric interaction that are present in the monomer 
 resolved simulations and prevent the unphysical crossing of chain 
 segments . in the coarse - grained simulations such a restriction is 
 absent ; i.e. , the effective rings are penetrable and can move apparently 
 through each other . on this level of description this effect is not 
 an unphysical process but should be interpreted as a shortcut connecting 
 initial and final configurations that are connected by a much more 
 time - consuming and physically realizable pathway of folding and collective 
 monomer movements . effective 
 center - of - mass pair potential in the isotropic effective 
 model for rings with different numbers of monomers n. the center - of - mass separation r is scaled with dg0 , the average diameter of gyration of a free 
 ring polymer . the solid line shows the angularly averaged effective 
 pair potential , while the dashed lines are results of ref ( 52 ) . we commence by recalculating the isotropic , i.e. , 
 angularly average , effective pair potential veffiso(r ) between the stiff rings as a way of comparison with the previously 
 derived results in ref ( 52 ) . the potential for r = 0 is 
 finite , as the rings are allowed to overlap . it also features a local 
 minimum there , whereas its maximum is located at r  0.25dg0 for all ring types 
 investigated . here , dg0 is the average 
 diameter of gyration of a free ring polymer . the height of the potential 
 barrier at small distances of r decreases by increasing 
 the number of monomers n on the ring polymers . infinite - dilution 
 limit of the quantity g(r , dd ) , which 
 quantifies the distribution of the scalar product 
 between directors for different values of r. we visualize 
 this distribution for the ring sizes n = 20 , n = 50 , and n = 100 . going over to the anisotropic effective model , we proceed 
 with 
 computing the aforementioned pair correlation function g(r , cos 1 , cos 2 ,  ) for a system of two ring polymers . as the latter depends 
 on four effective coordinates and is therefore difficult to visualize , 
 we first introduce a reduced pair correlation function g(r , dd ) , which expresses the relative joint probability density 
 of observing the two ring polymers at a distance r and with the directors mutually oriented at the value given by their 
 scalar product , dd , over the same quantity for noninteractive rings . moreover , 
 we introduce a reduced version of this function , g(r , dd ) , by dividing over its isotropic counterpart , i.e.12results are summarized 
 in figure 3 . one can 
 see that the angular 
 distribution of the directors changes significantly for r  0.25dg0 , which is approximately 
 the position of the maximum of veffiso(r ) . for r < 0.25dg0 the angle between 
 the directors is biased toward /2 , while for 0.25dg0 < r the position of the maximum of veffiso(r ) coincides 
 approximately with the distance r where interpenetrated 
 configurations of the rings become subdominant and where they are 
 more likely to align parallel to each other . the transition between 
 these two domains is particularly steep for n = 20 
 and becomes smoother for rings with a larger number of monomers . when 
 the rings interpenetrate each other , the distribution of angles between 
 the directors is rather wide , while it gets narrow after the transition 
 where the bias toward parallel alignments of the rings is very strong 
 in particular for the smallest rings with n = 20 . it is readily visible from figure 3 that anisotropy is particularly important for smaller 
 rings . for r > dg0 , the 
 distribution of the angle between the directors becomes flat , as the 
 rings are then well separated and hence do not interact . note that 
 by definition , eq 12 , the quantity g(r , dd ) , is a normalized 
 probability distribution for fixed r , and in figure 3 it is therefore 
 meaningless to compare the plotted function at different r values . effective potential for three different fixed configurations of 
 the directors and the connecting vector . as a comparison ( a ) n = 20 ; ( b ) n = 50 ; ( c ) n = 100 . the effective potentials are shown only for r values for which we have relatively good statistics . we also show 
 a sketch of the  ,  , and  
 configurations . the relative orientation 
 between the vectors r , d , 
 and d is of course 
 not completely determined by the scalar product dd ; the function g(r , dd ) contains less information than the full correlation 
 function g(r , cos 1 , cos 2 ,  ) . in particular , when the directors 
 are parallel , i.e. , dd = 1 , the angle between the connection vector r and the directors d and d is still arbitrary . we denote a configuration 
 with d||d||r as  and a configuration with d||dr as . from the reduced pair correlation function g(r , dd ) alone , we can not say which of these two configurations 
 is more probable , as the scalar product dd is identical to 1 in both cases . 
 using the full anisotropic pair correlation function g(r , cos 1 , cos 2 ,  ) , we can compare the corresponding effective potentials:13for the  case the 
 value of 
 the  coordinate is immaterial . however , due to the finite bin 
 size of the grid on which we have calculated g(r , cos 1 , cos 2 ,  ) , 
 the choice of  makes a small difference , even for v(r ) . we compute v(r ) from the 
 average of g(r , cos 1 = 1 , cos 2 = 1 ,  ) in . in figure 4 we see 
 that v(r ) increases significantly when r approaches dg0 , while v stays close to 0 until much smaller distances r. we can understand this results if we imagine the rings as disks 
 with diameter dg0 . in the  
 configuration , the rings lie in the same plane and will therefore 
 start to overlap as soon as r  dg0 . since the rings are not perfect circles and their 
 shape fluctuates , they can feel each other also for distances r which are slightly larger than dg0 . in the  configuration two disks overlap 
 only if the distance between their centers of mass is smaller than 
 their thickness . these results tell us that the peak in the reduced 
 pair correlation function g(r , dd ) for r  0.25dg0 and dd  
 1 is mostly due to  like configurations . however , 
 as soon as the rings can overlap in  type configurations 
 the effective potential increases very fast for smaller r , and we come to a regime where other configurations of the directors 
 are more favorable . as a comparison , we also consider a configuration 
 with dd||r , which we denote by . the corresponding 
 effective potential is given by14as in the  case , 
 the value 
 of the  coordinate is irrelevant for calculating v(r ) , which we compute 
 from the arithmetic mean of g(r , 
 cos 1 = 0 , cos 2 = 1 ,  ) 
 in . for small distances r one ring 
 interpenetrates 
 the other in microscopic configurations of type . 
 while v(r ) starts to increase at larger r values than v(r ) , the increase 
 is slower and converges to a constant for r  
 0 . this is intuitive to understand since it requires only a finite 
 amount of bending energy to deform two rings such that one can fit 
 into the other . the  becomes dominant over the  
 configuration at an r value below the threshold r  0.25dg0 . this is 
 also the r value at which we find the transition 
 in the reduced pair correlation function g(r , dd ) between a regime where configurations with parallel directors , 
 as in  , are preferred , to a regime where they are 
 suppressed and other configurations like  become dominant . first 
 coefficients in the expansion of g(r , cos 1 , cos 2 ,  ) 
 divided by the coefficient c0,0,0(r ) . ( a ) n = 20 , ( b ) n = 
 50 , and ( c ) n = 100 . in the appendix we explain how one 
 can 
 expand the angular part of g(r , 
 cos 1 , cos 2 ,  ) into a 
 series of suitably chosen basis functions fl1,l2,m(cos 1 , cos 2 ,  ) . the expression for this expansion is given in eq 27 , and the corresponding coefficients cl1,l2,m(r ) can be determined 
 by calculating particular ensemble averages as shown in eq 30 . we plot these coefficients cl1,l2,m(r ) for l1 , l2  2 
 in figure 5 . from the 
 fast change of cl1,l2,m(r ) for r 0.25dg0 one can once more see the transition between two regimes 
 for r in which the distribution of the directors 
 of the ring polymers is very different . the magnitude of coefficients cl1,l2,m(r)/c0,0,0(r ) with ( l1 , l2 )  ( 0 , 0 ) tells us 
 about the significance of the corresponding anisotropy in g(r , cos 1 , cos 2 ,  ) . anisotropy is more important for smaller rings 
 and becomes more pronounced after the transition at r  0.25dg0 , where the rings prefer 
 parallel configurations . pair correlation function g(r ) 
 at low reduced densities  * for a simulation of many ring 
 polymers in the full monomer - resolved simulation ( symbols ) , the anisotropic 
 effective model ( solid line ) , and the isotropic effective model ( dashed 
 line ) . we carried out monte carlo simulations 
 of systems of effective 
 particles described by the anisotropic effective model for different 
 ring sizes n and various densities . we define the 
 reduced density in our simulation as  *  ndg03/l , where n is the number of 
 rings in the sample . in order to assess the quality of the anisotropic 
 effective model , we compare our results to results of full monomer - resolved 
 simulations from ref ( 52 ) and the results of simulations using the isotropic effective model . as we can see in figure 6 for all choices of the number of monomers n the 
 effective models are in good agreement with the full monomer - resolved 
 simulations at low densities *. this is an important consistency 
 check for the effective models , in which the interactions have been 
 chosen such that the distribution of the effective degrees of freedom 
 agrees with their distribution in the full monomer - resolved simulations , 
 in the limit of small densities . pair correlation function g(r ) , at high densities , for a simulation of many 
 ring polymers with n = 20 monomers in the full monomer - resolved 
 simulation 
 ( symbols ) , the anisotropic effective model ( solid line ) , and the isotropic 
 effective model ( dashed line ) . in figure 7 , we 
 present results for the smallest rings with n = 20 
 monomers at higher densities . there is a dramatic improvement of the 
 accuracy as one compares the isotropic with the anisotropic model . while the former fails for  * > 2 the anisotropic effective 
 model works up to  *  5 and even gives a semiquantitatively 
 correct description of the system at  * = 5.97 . at the highest 
 densities , this peak in the pair correlation function is associated with the 
 emergence of stacks of parallel rings , and its position describes 
 the typical distance of rings in these stacks . interestingly , the isotropic effective potential has a maximum for r  0.25dg0 , which is close to the typical distance of the rings 
 in the stacks , and one could wonder why the rings prefer to align 
 at a distance which seems to have a very high free energy penalty 
 according to the isotropic effective potential . the answer to this 
 apparent paradox lies in the strongly peaked nature of the anisotropic 
 effective interaction , which we could observe in figures 3 , 4 , and 5 . while the average configuration of 
 the angular degrees of freedom at distances r  
 0.3dg0 has a high free energy penalty , 
 a certain class of configurations , where the directors of the rings 
 are almost parallel , is much more favorable . obviously , stacking can 
 not be observed in the isotropic effective model , where particles 
 possess no directional degrees of freedom . nr 
 in a simulation of many ring polymers with n = 20 
 monomers in the full monomer - resolved simulation ( symbols ) , the anisotropic 
 effective model ( solid line ) , and the isotropic effective model ( dashed 
 line ) . as a further characteristic of 
 the short - range coordination of 
 the rings , we consider the average number nr of neighbors within a distance r from the center of mass of a randomly chosen ring . this 
 is expressed as15for the rings with n = 20 
 monomers we present results for nr in figure 8 . once more , the good agreement between the full monomer - resolved 
 and the anisotropic effective model , even at the highest densities 
 investigated , is confirmed : small differences appear only for 0.25dg0  r  0.4dg0 . evidently , nr does not contain more information 
 than the g(r ) plot in figure 7 , but it nevertheless clarifies 
 the meaning of the disagreement between the g(r ) curves in the monomer - resolved and the anisotropic effective 
 model . at the highest densities in the full simulation , the centers 
 of mass move a bit closer to each other than they do in the anisotropic 
 effective simulation . this manifests itself as a shift of the peaks 
 in the g(r ) curves . the difference 
 in the height of the peaks is partly a consequence of the shift , since 
 a peak in g(r ) has to be higher 
 at smaller distances if it amounts to the same amount of average neighbors 
 as a peak at a larger distance r. the fact that the 
 nr curves 
 for the anisotropic effective and the full simulation in figure 8 agree for r  0.4dg0 shows us that 
 the peaks in the g(r ) curves indeed 
 correspond to the same amount of average nearby particles that are 
 simply accumulated at slightly different distances . pair correlation function g(r ) for a simulation of many ring polymers 
 with n = 
 50 monomers in the full monomer - resolved simulation ( symbols ) , the 
 anisotropic effective model ( solid line ) , and the isotropic effective 
 model ( dashed line ) . as can be seen by the pair correlation curves in figure 9a , also in this case the inclusion 
 of anisotropy improves the agreement with the monomer - resolved simulations 
 significantly for densities  * from 2.3 to 10.2 . in figure 9b we present g(r ) for higher *. in the full monomer - resolved 
 simulations we can see a peak emerging in the pair correlation function g(r ) at r = 0 on increasing 
 the density . as described in refs ( 52 and 58 ) , the monomer - resolved system forms stacks of quasi - parallel oblate 
 rings that are fully penetrated by bundles of elongated rings . in 
 this phase , the effective description , on the other hand , breaks down 
 if the internal configurations of the rings in the monomer - resolved 
 system differ significantly to the internal configurations in the 
 system with only two ring polymers . therefore , the anisotropic effective 
 model should not be expected to be a quantitative description at the 
 high densities in which this phase is formed . agreement with the monomer - resolved 
 model here is less satisfactory but the improvement over the isotropic 
 model is still spectacular . pair correlation function g(r ) for a simulation of many ring polymers with n = 
 100 monomers in the full monomer - resolved simulation ( symbols ) , the 
 anisotropic effective model ( solid line ) , and the isotropic effective 
 model ( dashed line ) . for the rings with n = 100 we find that anisotropy 
 does not play a key role any more , at least not for the full model 
 at the investigated densities . this had to be expected , as we could 
 already see in figure 3 and 5 that anisotropy is less pronounced 
 for larger ring sizes . as we saw in figure 6c , both the isotropic and the anisotropic 
 model give good results for g(r ) 
 up to  *  2.5 . in figure 10 , we see that for higher densities the inclusion 
 of anisotropy does not yield results that are in better agreement 
 with the full monomer - resolved simulations . the results in the isotropic 
 effective model even seem to be in better agreement with the full 
 model , which is attributed to multiparticle interactions that can 
 change the configurations of the large and therefore more deformable 
 rings significantly . the already small correlation between the directors , 
 which is present in the dilute case , might therefore be even smaller 
 at high densities . in the anisotropic effective model , we then overestimate 
 the angular correlations between the directors and arrive at results 
 that can be slightly worse than those of the isotropic model . interestingly , 
 at  * = 20.0 , which is the highest density investigated , the 
 anisotropic model appears to crystallize . at this density we see the 
 emergence of columns that are closed over the periodic boundary conditions 
 and organize in a hexagonal 2d lattice structure . p(dd ) is the 
 probability density to find the scalar product dd between 
 the directors of two close by rings ( r < 0.6dg0 ) . here we show p(dd ) for a simulation 
 of many ring polymers in the full monomer - resolved simulation ( symbols ) 
 and the anisotropic effective model ( solid line ) . ( a ) n = 20 , ( b ) n = 50 , and ( c ) n = 
 100 . we define p(dd ) as the probability density distribution for 
 the scalar products between the directors d and d of two ring polymers which are 
 a distance r < 0.6dg0 away from each other . in figure 11 , we present results for p(dd ) for simulations 
 in the monomer resolved and in the anisotropic effective model . if 
 the directors were uncorrelated , p(dd ) would be equal 
 to 1 . for low densities  * we obtain good agreement for all 
 ring sizes investigated . since we only look at the directional correlation 
 of close by ring polymers , p(dd ) can show strong anisotropic features 
 even for  *  0 . as expected , the anisotropy in p(dd ) is stronger 
 for smaller rings . when the density is increased , the distribution 
 always shifts toward parallel configurations in the effective model . this happens because less volume per ring is available for higher 
  * and by aligning parallel the rings occupy less space . typically 
 one observes the same trend in the monomer - resolved simulation ; only 
 for the n = 100 rings we find more parallel rings 
 for  * = 2.5 than for  * = 17.0 . in contrast to the effective 
 model the rings in the monomer - resolved simulation can deform , and 
 their interaction with other rings can therefore be more isotropic 
 at higher densities *. this explains why for the large rings 
 with n = 100 monomers , which deform more easily than 
 the smaller rings , the correlation between the directors is much weaker 
 than in the effective model and can even decrease with density . for n = 50 one can see that the number of orthogonal rings in 
 the monomer - resolved model at high densities is significantly larger 
 than in the effective simulation . as described in refs ( 52 and 58 ) for n = 50 and 
  * since the directors of the oblate and 
 the interpenetrating prolate rings can be orthogonal to each other , 
 one observes perpendicular directors for n = 50 even 
 at the highest densities investigated . in the anisotropic effective 
 model , on the other hand , this interpenetration is disfavored , and 
 we observe almost no orthogonal close - by rings at  * = 20 for n = 50 . pair correlation function g(r ) for a simulation of many ring polymers in the anisotropic effective 
 model . for the dashed line we expanded the pair - correlation function g before computing the associated effective pair potential . 
 for the expansion we took the 14 coefficients for which l1 , l2  4 into account . 
 the solid line shows results of a simulation with the unexpanded effective 
 pair interaction . instead of working with a fully tabulated effective 
 potential on 
 a four - dimensional grid , it can be advantageous to use the analytical 
 expansion on basis functions presented in the appendix . such expansions are truncated after some term , and here we shortly 
 discuss the quality of such truncations for the problem at hand . to 
 test the quality of the expansion of g(r , cos 1 , cos 2 ,  ) we also 
 carried out monte carlo simulations , where we used the effective potential 
 associated with the expanded correlation function as the pair interaction 
 between our effective particles . we took the 14 coefficients cl1,l2,m(r ) for which l1 , l2  4 
 into account and truncated the rest of the expansion . while g(r , cos 1 , cos 2 ,  ) can never be negative , the truncated expanded version 
 of g can accidentally become smaller than zero . wherever this happens g = 0 and veff =  are 
 used in the simulation . in figure 12 the pair correlation function g(r ) obtained in this simulation is shown in comparison with 
 the g(r ) function , which we computed 
 previously employing the full anisotropic effective potential . for 
 both n = 20 and n = 50 we obtain 
 reasonable results with the truncated effective interaction , given 
 by only 14 expansion coefficients . for the full effective interaction , 
 which we store on a 4d grid , we save 16 = 4096 entries 
 for each value of r ( see section 3.1 ) . at intermediate densities the results 
 obtained with the expanded effective interaction are a significant 
 improvement with respect to the isotropic effective model . however , 
 one has to be aware that for n = 20 the coefficients 
 of higher order modes can still be quite high , especially for r between 0.2dg0 and 0.7dg0 . in figure 5 we see that the coefficient for the mode with ( l1 , l2 , m ) = ( 0 , 2 , 0 ) can be larger than the coefficient of the isotropic 
 expansion mode . the reason for the high contribution of higher order 
 modes for n = 20 is of course the strongly peaked 
 nature of g(r , cos 1 , cos 2 ,  ) for these small rings , which 
 we can also observe in figure 3 . the convergence of g(r , cos 1 , cos 2 ,  ) is poor 
 for the n = 20 rings due to the strong anisotropy 
 of their effective interaction . however , our results show that the 
 expansion modes up to l1 , l2  4 already capture the main features of the 
 effective interaction . for n = 50 the degree of anisotropy 
 is weaker , and therefore the convergence of the expansion of g(r , cos 1 , cos 2 ,  ) is better . we have 
 introduced a minimal anisotropic model to coarse - grain 
 ring polymers with a finite bending rigidity as soft , penetrable disks . 
 for the shortest ( n = 20 ) and the intermediate ( n = 50 ) sized rings , this model represents a dramatic improvement 
 over the isotropic coarse - graining , in which the relative orientations 
 between the rings are all integrated upon and a radially symmetric 
 interaction results instead . the approach is capable of distinguishing 
 between the relative orientations at infinite dilution , and it carries 
 this distinction also to highly concentrated systems , where it reproduces 
 well the salient features of the structure as seen in the full monomer - resolved 
 simulations . whereas this is valid more for n = 20 
 and n = 50 , which have a contour length to persistence 
 length ratio of n / sp  2.7 and 6.7 , respectively , some important 
 features , such as the penetration of elongated rings in columns formed 
 by oblate rings ( found for n = 50 ) , are suppressed 
 or even lost in the effective description , as genuine many - body effects 
 come into play . for the largest rings , n = 100 , for 
 which we obtain n / sp  13.3 , the contour length is much larger 
 than the persistence length , and they thus resemble more flexible 
 objects . in this case this indeed reflects 
 the fact that such rings undergo , at high concentrations , conformational 
 changes ( shrinking , interpenetration ) that are quite distinct from 
 the assumptions that go into the anisotropic , soft disk model , rendering 
 it thereby very inaccurate . we therefore expect that our anisotropic 
 model yields quantitative results over a broad density range , for 
 systems of polymer rings with a contour to persistence length ratio 
 of n / sp  10 . our work provides , thus , an accurate and efficient 
 general scheme 
 for the coarse - graining of semiflexible ring polymers , as it allows 
 for a very dramatic reduction of their degrees of freedom , while at 
 the same time introducing a realistic class of systems for which anisotropic 
 generalizations of the ultrasoft , penetrable effective interactions 
 are physically meaningful . future work will focus on the investigations 
 of the structural and phase behavior of mixtures of stiff rings and 
 of the dynamics of the structure formation in the same .
we derive and introduce anisotropic 
 effective pair potentials to 
 coarse - grain solutions of semiflexible ring polymers of various lengths . 
 the system has been recently investigated by means of full monomer - resolved 
 computer simulations , revealing a host of unusual features and structure 
 formation , which , however , can not be captured by a rotationally averaged 
 effective pair potential between the rings centers of mass 
 [ bernabeim . ; 
 soft matter2013 , 9 , 1287 ] . 
 our new coarse - graining strategy is to picture each ring as a 
 soft , penetrable disk . 
 we demonstrate that for the short- and intermediate - length 
 rings the new model is quite capable of capturing the physics in a 
 quantitative fashion , whereas for the largest rings , which resemble 
 flexible ones , it fails at high densities . 
 our work opens the way 
 for the physical justification of general , anisotropic penetrable 
 interaction potentials .