Pressure_NN1 continues_VVZ to_TO0 be_VBI a_AT0 topic_NN1 of_PRF fundamental_AJ0 interest_NN1 in_PRP active_AJ0 matter_NN1 ._SENT An_AT0 interest_NN1 motivated_VVN ,_PUN in_PRP part_NN1 ,_PUN by_PRP the_AT0 tantalizing_AJ0 prospect_NN1 that_CJT mechanical_AJ0 equations-of-state_NN1 (_PUL EoS_NP0 )_PUR will_VM0 play_VVI as_AV0 important_AJ0 of_PRF a_AT0 role_NN1 in_PRP active_AJ0 matter_NN1 as_CJS they_PNP do_VDB in_PRP equilibrium_NN1 theory_NN1 ._SENT In_PRP fact_NN1 ,_PUN pressure_NN1 has_VHZ already_AV0 proven_VVN to_TO0 be_VBI a_AT0 salient_AJ0 metric_AJ0 for_PRP rationalizing_VVG the_AT0 often-complex_AJ0 behavior_NN1 of_PRF active_AJ0 suspensions_NN2 ranging_VVG from_PRP instabilities_NN2 exhibited_VVN by_PRP expanding_VVG bacterial_AJ0 droplets_NN2 ,_PUN vesicles_NN2 filled_VVN with_PRP active_AJ0 particles_NN2 ,_PUN active_AJ0 depletion_NN1 ,_PUN and_CJC the_AT0 dynamics_NN0 of_PRF colloidal_AJ0 gels_NN2 ,_PUN membranes_NN2 ,_PUN and_CJC polymers_NN2 immersed_VVN in_PRP a_AT0 bath_NN1 of_PRF active_AJ0 colloids_NN2 ._SENT The_AT0 burgeoning_NN1 of_PRF research_NN1 on_PRP the_AT0 nature_NN1 of_PRF pressure_NN1 in_PRP active_AJ0 systems_NN2 has_VHZ not_XX0 only_AV0 contributed_VVN to_TO0 progress_VVI in_PRP understanding_VVG the_AT0 phenomenology_NN1 of_PRF active_AJ0 systems_NN2 ,_PUN but_CJC has_VHZ also_AV0 played_VVN a_AT0 central_AJ0 role_NN1 in_PRP assessing_VVG the_AT0 validity_NN1 of_PRF new_AJ0 theoretical_AJ0 concepts_NN2 for_PRP nonequilibrium_AJ0 systems_NN2 ._SENT There_EX0 is_VBZ no_AT0 better_AJC example_NN1 than_CJS the_AT0 intense_AJ0 focus_NN1 on_PRP the_AT0 development_NN1 of_PRF nonequilibrium_AJ0 theories_NN2 for_PRP the_AT0 phase_NN1 behavior_NN1 of_PRF active_AJ0 particles_NN2 –_PUN a_AT0 crucial_AJ0 ingredient_NN1 for_PRP many_DT0 is_VBZ an_AT0 EoS_NP0 ._SENT 
Pressure_NN1 in_PRP active_AJ0 systems_NN2 remains_VVZ difficult_AJ0 to_TO0 characterize_VVI analytically_AV0 in_PRP all_DT0 but_CJC the_AT0 simplest_AJS cases_NN2 ._SENT These_DT0 challenges_NN2 can_VM0 be_VBI attributed_VVN to_PRP a_AT0 unique_AJ0 nonlocal_AJ0 contribution_NN1 to_PRP the_AT0 pressure_NN1 referred_VVN to_PRP as_PRP the_AT0 swim_NN1 pressure_NN1 ._SENT This_DT0 nontraditional_AJ0 source_NN1 of_PRF stress_NN1 is_VBZ entirely_AV0 rooted_VVN in_PRP trajectory_NN1 space_NN1 and_CJC unraveling_VVG its_DPS concentration_NN1 dependence_NN1 necessitates_VVZ the_AT0 difficult_AJ0 task_NN1 of_PRF understanding_NN1 how_AVQ particle_NN1 trajectories_NN2 are_VBB altered_VVN by_PRP many-body_AJ0 correlations_NN2 ._SENT The_AT0 driving_AJ0 force_NN1 for_PRP much_DT0 of_PRF the_AT0 novel_AJ0 behaviors_NN2 observed_VVN in_PRP active_AJ0 systems_NN2 ,_PUN including_PRP the_AT0 phenomena_NN2 of_PRF motility-induced_AJ0 phase_NN1 separation_NN1 (_PUL MIPS_NN0 )_PUR ,_PUN is_VBZ due_AJ0 to_PRP this_DT0 inherently_AV0 nonequilibrium_AJ0 contribution_NN1 to_PRP the_AT0 pressure_NN1 ._SENT 
The_AT0 swim_NN1 pressure_NN1 emerges_VVZ from_PRP the_AT0 persistent_AJ0 and_CJC time-irreversible_AJ0 motion_NN1 generated_VVN by_PRP the_AT0 non-conservative_AJ0 self-propelling_AJ0 force_NN1 of_PRF active_AJ0 particles_NN2 ._SENT Within_PRP the_AT0 context_NN1 of_PRF the_AT0 active_AJ0 Brownian_AJ0 particle_NN1 (_PUL ABP_NP0 )_PUR model_NN1 a_AT0 popular_AJ0 minimal_AJ0 model_NN1 for_PRP studying_VVG active_AJ0 colloids_NN2 particles_NN2 self-propel_NN1 at_PRP a_AT0 constant_AJ0 speed_NN1 U0_NN1 and_CJC undergo_VVB Brownian_AJ0 rotational_AJ0 diffusion_NN1 with_PRP a_AT0 characteristic_AJ0 reorientation_NN1 time_NN1 τR_NN1 ._SENT At_PRP long_AJ0 times_NN2 ,_PUN an_AT0 ideal_AJ0 ABP_NN1 executes_VVZ a_AT0 random_AJ0 walk_NN1 with_PRP a_AT0 trajectory_NN1 correlation_NN1 length_NN1 l0_NP0 =_NP0 U0_NP0 τR_NP0 ,_PUN which_DTQ we_PNP call_VVB the_AT0 intrinsic_AJ0 run-length_NN1 ._SENT A_AT0 dilute_AJ0 system_NN1 of_PRF ABPs_NN2 with_PRP number_NN1 density_NN1 n_ZZ0 exerts_VVZ a_AT0 swim_NN1 pressure_NN1 directly_AV0 proportional_AJ0 to_PRP this_DT0 in-trinsic_AJ0 run-length_NN1 ,_PUN equation_NN1 ._SENT For_PRP more_DT0 concentrated_AJ0 systems_NN2 ,_PUN interparticle_NN1 interactions_NN2 reduce_VVB the_AT0 tra_NN1 jectory_NN1 correlation_NN1 length_NN1 ,_PUN resulting_VVG in_PRP a_AT0 density-dependent_AJ0 effective_AJ0 run-length_NN1 l_ZZ0 <_NN1 l0_NN1 and_CJC a_AT0 swim_NN1 pressure_NN1 of_PRF Πs_NP0 ∼_NP0 nlU0_NP0 ._SENT In_PRP the_AT0 limit_NN1 of_PRF maximal-packing_NN1 ,_PUN ABPs_NN2 are_VBB prevented_VVN from_PRP executing_VVG their_DPS athermal_AJ0 random_AJ0 walk_NN1 resulting_VVG in_PRP a_AT0 vanishingly_AV0 small_AJ0 effective_AJ0 run-length_NN1 (_PUL and_CJC swim_NN1 pressure_NN1 )_PUR that_DT0 should_VM0 be_VBI independent_AJ0 of_PRF l0_NN1 and_CJC depend_VVB only_AV0 on_PRP geometric_AJ0 packing_NN1 constraints_NN2 ._SENT This_DT0 physical_AJ0 expectation_NN1 for_PRP the_AT0 swim_NN1 pressure_NN1 is_VBZ currently_AV0 not_XX0 reflected_VVN in_PRP the_AT0 literature_NN1 ._SENT 
This_DT0 simple_AJ0 physical_AJ0 picture_NN1 [_PUL see_VVB Fig._NN1 1_CRD ]_PUR suggests_VVZ the_AT0 existence_NN1 of_PRF a_AT0 concentration_NN1 scale_NN1 that_CJT controls_VVZ the_AT0 crossover_NN1 between_PRP the_AT0 dynamically_AV0 distinct_AJ0 dilute_AJ0 and_CJC jammed_AJ0 regimes_NN2 of_PRF active_AJ0 suspensions_NN2 ._SENT In_PRP this_DT0 Letter_NN1 ,_PUN using_VVG large-scale_AJ0 computer_NN1 simulations_NN2 ,_PUN we_PNP identify_VVB this_DT0 new_AJ0 run-length_AJ0 dependent_AJ0 concentration_NN1 φ∗_NN1 ,_PUN which_DTQ exhibits_VVZ a_AT0 number_NN1 of_PRF compelling_AJ0 analogies_NN2 to_PRP the_AT0 powerful_AJ0 concept_NN1 of_PRF the_AT0 overlap_NN1 concentration_NN1 of_PRF equilibrium_NN1 polymer_NN1 solutions_NN2 ._SENT Using_VVG φ∗_NN1 and_CJC concentration_NN1 scales_NN2 associated_VVN with_PRP colloidal_AJ0 jamming_NN1 ,_PUN the_AT0 mechanical_AJ0 equation-of-state_NN1 for_PRP ABPs_NN2 can_VM0 be_VBI collapsed_VVN onto_PRP a_AT0 single_AJ0 curve_NN1 which_DTQ contains_VVZ a_AT0 number_NN1 of_PRF previously_AV0 overlooked_VVN features_NN2 ,_PUN the_AT0 inclusion_NN1 of_PRF which_DTQ qualitatively_AV0 alters_VVZ previous_AJ0 predictions_NN2 of_PRF the_AT0 phase_NN1 behavior_NN1 of_PRF ABPs_NN2 ._SENT Moreover_AV0 ,_PUN the_AT0 use_NN1 of_PRF a_AT0 dynamic_AJ0 overlap_NN1 concentration_NN1 scale_NN1 may_VM0 prove_VVI to_TO0 be_VBI of_PRF great_AJ0 utility_NN1 in_PRP the_AT0 construction_NN1 of_PRF accurate_AJ0 nonequilibrium_NN1 equations-of-state_NN1 ._SENT 
Simulation_NN1 Model_NN1 and_CJC the_AT0 EoS_NP0 ._SENT –_PUN In_PRP an_AT0 effort_NN1 to_TO0 unpack_VVI the_AT0 concentration_NN1 dependence_NN1 of_PRF the_AT0 pressure_NN1 ,_PUN we_PNP focus_VVB on_PRP the_AT0 simplest_AJS and_CJC most_AV0 widely_AV0 studied_VVN active_AJ0 system_NN1 :_PUN purely-repulsive_AJ0 active_AJ0 Brownian_AJ0 disks_NN2 ._SENT By_PRP introducing_VVG a_AT0 small_AJ0 degree_NN1 of_PRF polydispersity_NN1 in_PRP disk_NN1 size_NN1 ,_PUN we_PNP ensure_VVB the_AT0 system_NN1 remains_VVZ disordered_AJ0 for_PRP all_DT0 concentrations_NN2 ._SENT This_DT0 is_VBZ in_PRP contrast_NN1 to_PRP monodisperse_NN1 active_AJ0 disks_NN2 ,_PUN which_DTQ exhibit_VVB a_AT0 order-disorder_NN1 phase_NN1 transition_NN1 with_PRP a_AT0 complex_AJ0 dependence_NN1 on_PRP activity_NN1 ._SENT The_AT0 small_AJ0 degree_NN1 of_PRF polydispersity_NN1 allows_VVZ us_PNP to_TO0 isolate_VVI the_AT0 disordered_AJ0 branch_NN1 of_PRF the_AT0 equations-of-state_NN1 ,_PUN which_DTQ not_XX0 only_AV0 will_VM0 serve_VVI as_PRP a_AT0 useful_AJ0 reference_NN1 system_NN1 for_PRP active_AJ0 fluids_NN2 with_PRP more_DT0 complicated_AJ0 interactions_NN2 ,_PUN but_CJC allows_VVZ for_PRP the_AT0 quantitative_AJ0 study_NN1 of_PRF the_AT0 interplay_NN1 between_PRP activity_NN1 and_CJC concentration_NN1 ._SENT 
We_PNP consider_VVB a_AT0 two_CRD dimensional_AJ0 system_NN1 of_PRF overdamped_AJ0 disks_NN2 at_PRP fixed_AJ0 number_NN1 density_NN1 n_ZZ0 where_AVQ each_DT0 particle_NN1 experiences_VVZ a_AT0 drag_NN1 force_NN1 ζ_NP0 U_NP0 ,_PUN a_AT0 conservative_AJ0 interparticle_NN1 force_NN1 F_ZZ0 c_ZZ0 and_CJC an_AT0 active_AJ0 self-propelling_NN1 force_NN1 F_ZZ0 a_AT0 =_NN1 ζU0q_NN1 ._SENT The_AT0 orientation_NN1 of_PRF each_DT0 particle_NN1 q_ZZ0 is_VBZ independent_AJ0 and_CJC obeys_VVZ diffusive_AJ0 rotary_AJ0 dynamics_NN0 with_PRP a_AT0 characteristic_AJ0 reorientation_NN1 time_NN1 τR_NN1 ._SENT The_AT0 sum_NN1 of_PRF the_AT0 forces_NN2 results_NN2 in_PRP a_AT0 simple_AJ0 equation-of-motion_NN1 for_PRP the_AT0 particle_NN1 velocity_NN1 :_PUN equation_NN1 ._SENT The_AT0 interparticle_NN1 force_NN1 F_ZZ0 c_ZZ0 arises_VVZ from_PRP a_AT0 Weeks-Chandler-Anderson_NP0 (_PUL WCA_NP0 )_PUR potential_NN1 characterized_VVN by_PRP a_AT0 potential_AJ0 depth_NN1 ε_NN1 and_CJC an_AT0 average_AJ0 Lennard-Jones_NP0 diameter_NN1 d_ZZ0 ._SENT Particle_NN1 diameters_NN2 are_VBB drawn_VVN from_PRP a_AT0 normal_AJ0 distribution_NN1 with_PRP a_AT0 standard_AJ0 deviation_NN1 of_PRF 0.1d_NN1 ._SENT Importantly_AV0 ,_PUN as_CJS our_DPS active_AJ0 force_NN1 is_VBZ of_PRF finite_AJ0 amplitude_NN1 ,_PUN a_AT0 sufficiently_AV0 strong_AJ0 choice_NN1 for_PRP the_AT0 repulsive_AJ0 force_NN1 F_ZZ0 c_ZZ0 will_VM0 mimic_VVI a_AT0 true_AJ0 hard-particle_AJ0 potential_NN1 ._SENT A_AT0 choice_NN1 of_PRF ε_NN1 /_PUN (_PUL ζU0d_AJ0 )_PUR =_NN1 100_CRD is_VBZ found_VVN to_TO0 result_VVI in_PRP hard_AJ0 disk_NN1 statistics_NN0 with_PRP an_AT0 effective_AJ0 average_AJ0 particle_NN1 diameter_NN1 of_PRF 21_CRD /_PUN 6d_NN0 ._SENT In_PRP this_DT0 hard-disk_AJ0 limit_NN1 ,_PUN the_AT0 state_NN1 of_PRF our_DPS system_NN1 is_VBZ independent_AJ0 of_PRF the_AT0 amplitude_NN1 of_PRF the_AT0 active_AJ0 force_NN1 and_CJC is_VBZ fully_AV0 described_VVN by_PRP two_CRD geometric_AJ0 parameters_NN2 :_PUN the_AT0 area_NN1 fraction_NN1 φ_NP0 =_NP0 nπ_NP0 (_PUL 21_CRD /_PUN 6d_NN0 )_PUR 2_CRD /_PUN 4_CRD and_CJC the_AT0 dimensionless_AJ0 intrinsic_AJ0 run-length_NN1 l0_CRD /_PUN d_ZZ0 ._SENT Using_VVG the_AT0 GPU-enabled_AJ0 HOOMD-blue_AJ0 software_NN1 package_NN1 ,_PUN all_DT0 simulations_NN2 were_VBD conducted_VVN with_PRP 40_CRD ,_PUN 000_CRD particles_NN2 and_CJC run_VVN for_PRP a_AT0 minimum_AJ0 duration_NN1 of_PRF 5_CRD ,_PUN 000_CRD d_ZZ0 /_PUN U0_CRD ._SENT 
The_AT0 total_AJ0 mechanical_AJ0 pressure_NN1 Π_NN1 has_VHZ two_CRD contributions_NN2 :_PUN the_AT0 collisional_AJ0 pressure_NN1 Πc_NN1 arising_VVG from_PRP conservative_AJ0 interparticle_NN1 interactions_NN2 and_CJC the_AT0 swim_NN1 pressure_NN1 Πs_NN2 generated_VVN by_PRP the_AT0 active_AJ0 force_NN1 ._SENT The_AT0 collisional_AJ0 pressure_NN1 follows_VVZ from_PRP the_AT0 standard_AJ0 micromechanical_AJ0 virial_NN1 for_PRP conservative_AJ0 interactions_NN2 Πc_NP0 =_NP0 n⟨x_NP0 ·_NN0 F_ZZ0 c_ZZ0 ⟩_NN1 /_PUN 2_CRD where_AVQ x_ZZ0 is_VBZ the_AT0 particle_NN1 position_NN1 and_CJC ⟨_NN1 ..._PUN ⟩_NP0 denotes_VVZ an_AT0 average_NN1 over_PRP all_DT0 particles_NN2 ._SENT At_PRP steady-state_AJ0 ,_PUN Πs_NN2 can_VM0 be_VBI written_VVN in_PRP the_AT0 impulse_NN1 form_NN1 :_PUN equation_NN1 ._SENT For_PRP an_AT0 isotropic_AJ0 system_NN1 of_PRF active_AJ0 particles_NN2 free_AJ0 of_PRF aligning_VVG interaction_NN1 ,_PUN Eq._NP0 (_PUL 2_CRD )_PUR is_VBZ the_AT0 pressure_NN1 that_CJT the_AT0 suspension_NN1 would_VM0 exert_VVI on_PRP a_AT0 flat_AJ0 torque-free_AJ0 macroscopic_AJ0 boundary_NN1 ._SENT Under_PRP these_DT0 conditions_NN2 ,_PUN the_AT0 total_AJ0 pressure_NN1 exerted_VVN on_PRP the_AT0 surroundings_NN2 is_VBZ simply_AV0 the_AT0 sum_NN1 of_PRF the_AT0 collisional_AJ0 and_CJC swim_NN1 pressure_NN1 Π_NP0 =_NP0 Πc_NP0 +_NP0 Πs_NP0 ._SENT 
Equation_NN1 (_PUL 2_CRD )_PUR allows_VVZ us_PNP to_TO0 directly_AV0 probe_VVI via_PRP simulation_NN1 the_AT0 effective_AJ0 run-length_NN1 of_PRF the_AT0 particles_NN2 ,_PUN l_ZZ0 ≡_NP0 τR_NP0 ⟨q_NP0 ·_NN0 U_ZZ0 ⟩_NN1 ,_PUN which_DTQ is_VBZ the_AT0 true_AJ0 correlation_NN1 length_NN1 of_PRF a_AT0 particle_NN1 's_POS trajectory_NN1 and_CJC a_AT0 measure_NN1 of_PRF the_AT0 correlation_NN1 between_PRP the_AT0 orientation_NN1 of_PRF a_AT0 particle_NN1 and_CJC the_AT0 forces_NN2 (_PUL both_AV0 active_AJ0 and_CJC interparticle_AJ0 )_PUR acting_VVG upon_PRP it_PNP ._SENT Analytical_AJ0 expressions_NN2 for_PRP the_AT0 effective_AJ0 run-length_NN1 l_ZZ0 require_VVB solving_VVG a_AT0 many-body_AJ0 dynamics_NN0 problem_NN1 in_PRP both_DT0 position_NN1 and_CJC orientation_NN1 space_NN1 ,_PUN and_CJC have_VHB been_VBN derived_VVN in_PRP only_AV0 a_AT0 few_DT0 asymptotic_AJ0 limits_NN2 ._SENT Additionally_AV0 ,_PUN the_AT0 nature_NN1 of_PRF l_ZZ0 is_VBZ further_AV0 obfuscated_VVN as_PRP direct_AJ0 measurement_NN1 is_VBZ limited_VVN to_PRP a_AT0 narrow_AJ0 region_NN1 of_PRF (_PUL l0_CRD ,_PUN φ_NP0 )_PUR -state-space_NN1 where_CJS motility-induced_AJ0 phase_NN1 separation_NN1 is_VBZ absent_AJ0 (_PUL i.e._AV0 ,_PUN outside_AV0 of_PRF the_AT0 MIPS_NN0 phase_NN1 envelope_NN1 )_PUR ._SENT Paradoxically_AV0 ,_PUN the_AT0 prediction_NN1 of_PRF both_DT0 the_AT0 spinodal_AJ0 and_CJC binodal_AJ0 requires_VVZ knowledge_NN1 of_PRF l_ZZ0 in_PRP regions_NN2 where_AVQ it_PNP cannot_VVB be_VBI directly_AV0 measured_VVN ._SENT Extrapolation_NN1 of_PRF l_ZZ0 into_PRP these_DT0 mechanically_AV0 forbidden_AJ0 regions_NN2 requires_VVZ a_AT0 complete_AJ0 understanding_NN1 of_PRF the_AT0 trends_NN2 in_PRP l_ZZ0 as_PRP the_AT0 system_NN1 approaches_VVZ the_AT0 critical_AJ0 l0_NN0 for_PRP MIPS_NN0 (_PUL l0_CRD /_PUN d_ZZ0 ≈_NN1 13_CRD )_PUR ._SENT Figure_NN1 2_CRD (_PUL a_AT0 )_PUR presents_VVZ the_AT0 complete_AJ0 φ_NN1 dependence_NN1 of_PRF l_ZZ0 in_PRP regions_NN2 of_PRF (_PUL l0_CRD ,_PUN φ_NP0 )_PUR -statespace_NN1 where_AVQ the_AT0 system_NN1 is_VBZ homogeneous_AJ0 ._SENT 
Our_DPS approach_NN1 for_PRP isolating_VVG these_DT0 trends_NN2 is_VBZ motivated_VVN by_PRP two_CRD simple_AJ0 observations_NN2 ._SENT First_ORD ,_PUN l_ZZ0 becomes_VVZ increasingly_AV0 independent_AJ0 of_PRF the_AT0 intrinsic_AJ0 run-length_AJ0 l0_NN1 with_PRP increasing_AJ0 concentration_NN1 ._SENT In_PRP the_AT0 dilute_AJ0 limit_NN1 ,_PUN l_ZZ0 =_NP0 l0_NP0 =_NP0 U0_NP0 τR_NP0 ,_PUN while_CJS in_PRP the_AT0 limit_NN1 of_PRF maximal-packing_NN1 l_ZZ0 ≈_NN1 0._CRD Second_ORD ,_PUN deviations_NN2 from_PRP the_AT0 dilute_AJ0 scaling_NN1 prediction_NN1 of_PRF l_ZZ0 occur_VVB at_PRP lower_AJC concentrations_NN2 as_CJS l0_NN1 is_VBZ increased_VVN ._SENT Microrheology_NN1 calculations_NN2 for_PRP hard-disks_NN2 show_VVB the_AT0 first-order_AJ0 correction_NN1 to_PRP the_AT0 effective_AJ0 run-length_NN1 is_VBZ given_VVN by_PRP l_CRD ≈_NP0 l0_CRD (_PUL 1_CRD −_NP0 φ_NP0 )_PUR in_PRP the_AT0 limit_NN1 of_PRF large_AJ0 l0_NN1 ._SENT As_CJS shown_VVN in_PRP Fig._NN1 2_CRD (_PUL b_ZZ0 )_PUR ,_PUN this_DT0 expression_NN1 is_VBZ in_PRP excellent_AJ0 agreement_NN1 with_PRP our_DPS data_NN0 for_PRP persistent_AJ0 ABPs_NN2 (_PUL l0_CRD >_NN0 3d_NN0 )_PUR ,_PUN and_CJC note_VVB the_AT0 previously_AV0 mentioned_VVN trend_NN1 in_PRP the_AT0 deviation_NN1 from_PRP this_DT0 dilute_AJ0 scaling_NN1 as_PRP l0_CRD increases_NN2 ._SENT 
Dynamic_AJ0 Overlap_NN1 Concentration_NN1 ._SENT –_PUN The_AT0 dependence_NN1 of_PRF the_AT0 swim_NN1 pressure_NN1 on_PRP a_AT0 single-particle_AJ0 length_NN1 scale_NN1 at_PRP low_AJ0 concentrations_NN2 –_PUN the_AT0 intrinsic_AJ0 run-length_NN1 l0_NN1 –_PUN and_CJC its_DPS independence_NN1 from_PRP that_DT0 length_NN1 scale_NN1 at_PRP high_AJ0 concentrations_NN2 suggests_VVZ a_AT0 connection_NN1 to_PRP equilibrium_NN1 polymer_NN1 solutions_NN2 ._SENT There_AV0 exist_VVB distinct_AJ0 concentration_NN1 regimes_NN2 for_PRP polymer_NN1 solutions_NN2 ,_PUN corresponding_VVG to_PRP isolated_AJ0 (_PUL dilute_AJ0 )_PUR and_CJC overlapping_AJ0 chains_NN2 (_PUL semidilute_AJ0 )_PUR ._SENT The_AT0 thermodynamic_AJ0 equations-of-state_NN1 in_PRP each_DT0 regime_NN1 have_VHB distinct_AJ0 dependencies_NN2 on_PRP single-chain_AJ0 properties_NN2 ._SENT For_PRP example_NN1 ,_PUN the_AT0 osmotic_AJ0 pressure_NN1 of_PRF a_AT0 dilute_AJ0 polymer_NN1 solution_NN1 is_VBZ inversely_AV0 proportional_AJ0 to_PRP the_AT0 degree_NN1 of_PRF polymerization_NN1 N_ZZ0 [_PUL see_VVB Fig._NN1 1_CRD ]_PUR ._SENT In_PRP the_AT0 semidilute_AJ0 regime_NN1 where_AVQ chains_NN2 strongly_AV0 overlap_VVB ,_PUN the_AT0 individuality_NN1 of_PRF each_DT0 chain_NN1 is_VBZ lost_VVN resulting_VVG in_PRP an_AT0 osmotic_AJ0 pressure_NN1 that_DT0 is_VBZ independent_AJ0 of_PRF the_AT0 single-chain_AJ0 property_NN1 N_ZZ0 ._SENT The_AT0 overlap_NN1 concentration_NN1 φ∗_NN1 delineates_VVZ the_AT0 dilute_AJ0 and_CJC semidilute_AJ0 regimes_NN2 and_CJC is_VBZ the_AT0 concentration_NN1 at_PRP which_DTQ polymer_NN1 coils_NN2 begin_VVB to_TO0 overlap_VVI ._SENT The_AT0 density_NN1 of_PRF chains_NN2 at_PRP φ∗_NP0 is_VBZ inversely_AV0 proportional_AJ0 to_PRP the_AT0 effective_AJ0 volume_NN1 occupied_VVN by_PRP a_AT0 single_AJ0 polymer_NN1 chain_NN1 V_ZZ0 chain_NN1 ._SENT 
For_PRP ABPs_NN2 ,_PUN we_PNP seek_VVB the_AT0 characteristic_AJ0 single-particle_AJ0 volume_NN1 (_PUL area_NN1 in_PRP 2D_NN0 )_PUR VABP_NN1 that_CJT plays_VVZ the_AT0 analogous_AJ0 role_NN1 to_PRP the_AT0 single-chain_AJ0 volume_NN1 V_ZZ0 chain_NN1 for_PRP polymer_NN1 solutions_NN2 ._SENT In_PRP 2D_NN0 ,_PUN a_AT0 natural_AJ0 starting_NN1 point_NN1 is_VBZ to_TO0 consider_VVI the_AT0 area_NN1 swept_VVD out_AVP by_PRP an_AT0 ideal_AJ0 ABP_NN1 in_PRP a_AT0 single_AJ0 reorientation_NN1 time_NN1 τR_NN1 :_PUN V_ZZ0 ABP_NP0 ∼_NN1 l0d_NN1 ._SENT For_PRP concentrations_NN2 beyond_PRP φ∗_NP0 ∼_NP0 1_CRD /_PUN V_ZZ0 ABP_AJ0 particle_NN1 trajectories_NN2 overlap_NN1 and_CJC collisions_NN2 prevent_VVB ABPs_NN2 from_PRP executing_VVG their_DPS intrinsic_AJ0 random_AJ0 walk_NN1 of_PRF length_NN1 l0_CRD [_PUL see_VVB Fig._NN1 1_CRD ]_PUR ._SENT The_AT0 single-particle_AJ0 volume_NN1 given_VVN by_PRP V_CRD ABP_NN1 ∼_NN1 l0d_NN1 is_VBZ likely_AJ0 an_AT0 overestimate_NN1 of_PRF the_AT0 overlap_NN1 probability_NN1 ._SENT More_AV0 realistically_AV0 ,_PUN we_PNP anticipate_VVB a_AT0 weaker_AJC dependence_NN1 on_PRP l0_NN1 and_CJC introduce_VVB the_AT0 more_AV0 general_AJ0 definitionVABP_NN1 ∼lλ0d2−λ_NP0 where0_NP0 <_NP0 λ_NP0 <_NN1 1isaconstanttobe_NN1 determined_VVN ._SENT The_AT0 resulting_AJ0 overlap_NN1 concentration_NN1 φ∗_NN1 can_VM0 be_VBI written_VVN as_AV0 :_PUN equation_NN1 ._SENT 
The_AT0 dynamic_AJ0 overlap_NN1 concentration_NN1 φ∗_NN1 proposed_VVN in_PRP Eq._NP0 (_PUL 3_CRD )_PUR should_VM0 delineate_VVI dilute_AJ0 and_CJC semidilute_AJ0 concentration_NN1 regimes_NN2 for_PRP all_DT0 values_NN2 of_PRF l0_NN1 ._SENT Using_VVG this_DT0 criterion_NN1 as_PRP a_AT0 guide_NN1 (_PUL see_VVB Supplemental_AJ0 Material_NN1 )_PUR ,_PUN we_PNP identify_VVB a_AT0 significant_AJ0 dependence_NN1 of_PRF the_AT0 dynamic_AJ0 overlap_NN1 concentration_NN1 on_PRP the_AT0 intrinsic_AJ0 run-length_NN1 with_PRP λ_NP0 ≈_NP0 0.353_CRD for_PRP our_DPS polydisperse_NN1 system_NN1 of_PRF active_AJ0 Brownian_AJ0 disks_NN2 ._SENT As_CJS shown_VVN in_PRP Fig._NN1 2_CRD (_PUL c_ZZ0 )_PUR ,_PUN rescaling_VVG φ_NN1 with_PRP φ∗_NN1 allows_VVZ for_PRP a_AT0 collapse_NN1 of_PRF l_ZZ0 in_PRP the_AT0 dilute_AJ0 (_PUL φ_NP0 /_PUN φ∗_NP0 <_NP0 1_CRD )_PUR and_CJC semidilute_NN1 (_PUL φ_NP0 /_PUN φ∗_NP0 ≥_NP0 1_CRD )_PUR regimes_NN2 ._SENT Notably_AV0 ,_PUN for_PRP each_DT0 l0_CRD deviations_NN2 from_PRP the_AT0 collapse_NN1 begin_VVB to_TO0 occur_VVI at_PRP a_AT0 φ∗-independent_AJ0 area_NN1 fraction_NN1 φ_NP0 ≈_NP0 0.72_CRD ,_PUN indicative_AJ0 of_PRF an_AT0 upper-bound_AJ0 for_PRP our_DPS semidilute_AJ0 regime_NN1 and_CJC the_AT0 emergence_NN1 of_PRF what_DTQ we_PNP term_VVB as_PRP the_AT0 active_AJ0 jamming_NN1 regime_NN1 ._SENT 
As_CJS the_AT0 system_NN1 transitions_NN2 to_PRP the_AT0 active_AJ0 jamming_NN1 regime_NN1 ,_PUN the_AT0 run-length_NN1 tends_VVZ to_PRP zero_CRD as_PRP the_AT0 maximalpacking_AJ0 φm_NN1 ≈_NN1 0.845_CRD is_VBZ approached_VVN ._SENT Furthermore_AV0 ,_PUN the_AT0 run-length_NN1 becomes_VVZ increasingly_AV0 independent_AJ0 of_PRF l0_NN1 as_PRP geometric_AJ0 packing_NN1 constraints_NN2 set_VVB the_AT0 scale_NN1 for_PRP particle_NN1 motion_NN1 ._SENT We_PNP find_VVB l_ZZ0 in_PRP the_AT0 active_AJ0 jamming_NN1 regime_NN1 is_VBZ welldescribed_AJ0 by_PRP :_PUN equation_NN1 ,_PUN where_AVQ D_ZZ0 ,_PUN E_ZZ0 >_NN1 0_CRD are_VBB constant_AJ0 empirical_AJ0 fitting_AJ0 parameters_NN2 ._SENT As_CJS shown_VVN by_PRP the_AT0 family_NN1 of_PRF curves_NN2 in_PRP Fig._NN1 2_CRD (_PUL c_ZZ0 )_PUR ,_PUN Eq._NP0 (_PUL 4_CRD )_PUR accurately_AV0 captures_VVZ the_AT0 deviation_NN1 of_PRF l_ZZ0 from_PRP the_AT0 semidilute_AJ0 regime_NN1 as_PRP a_AT0 function_NN1 of_PRF l0_NN1 while_CJS also_AV0 capturing_VVG our_DPS physical_AJ0 asymptotic_AJ0 expectations_NN2 ._SENT With_PRP increasing_AJ0 l0_NN1 ,_PUN the_AT0 effective_AJ0 run-length_NN1 approaches_VVZ a_AT0 constant_AJ0 value_NN1 set_VVN by_PRP the_AT0 packing_NN1 geometry_NN1 i.e._AV0 ,_PUN liml0→∞_NP0 l_ZZ0 =_NP0 DE_NP0 (_PUL 1_CRD −_NP0 φ_NP0 /_PUN φm_NN1 )_PUR ._SENT Previous_AJ0 EoS_NP0 which_DTQ expressed_VVD l_CRD =_NP0 l0_CRD f_ZZ0 (_PUL φ_NP0 )_PUR (_PUL for_PRP large_AJ0 l0_NN1 )_PUR fail_VVB to_TO0 capture_VVI this_DT0 physical_AJ0 expectation_NN1 and_CJC ,_PUN in_PRP fact_NN1 ,_PUN find_VVB the_AT0 unphysical_AJ0 result_NN1 of_PRF liml0→∞_NN1 l_ZZ0 =_NN1 ∞_NN1 for_PRP all_DT0 φ_NN1 unless_CJS f_ZZ0 (_PUL φ_NP0 )_PUR is_VBZ identically_AV0 zero_CRD ._SENT 
By_PRP identifying_VVG φ∗_NN1 and_CJC the_AT0 scaling_NN1 behavior_NN1 in_PRP the_AT0 active_AJ0 jamming_NN1 regime_NN1 ,_PUN the_AT0 l0_AJ0 dependence_NN1 of_PRF the_AT0 three_CRD regions_NN2 of_PRF the_AT0 swim_NN1 pressure_NN1 (_PUL dilute_AJ0 ,_PUN semidilute_AJ0 and_CJC active_AJ0 jamming_NN1 )_PUR can_VM0 be_VBI described_VVN quantitatively_AV0 ._SENT From_PRP this_DT0 information_NN1 ,_PUN it_PNP is_VBZ straight-forward_AJ0 to_TO0 construct_VVI a_AT0 functional_AJ0 form_NN1 that_CJT captures_VVZ the_AT0 full_AJ0 φ_NN1 and_CJC l0_AJ0 dependencies_NN2 of_PRF the_AT0 swim_NN1 pressure_NN1 ._SENT In_PRP Fig._NN1 2_CRD (_PUL d_ZZ0 )_PUR ,_PUN representative_AJ0 curves_NN2 for_PRP the_AT0 φ∗-informed_AJ0 swim_NN1 pressure_NN1 are_VBB provided_VVN for_PRP a_AT0 range_NN1 of_PRF l0_NN1 above_PRP the_AT0 critical_AJ0 intrinsic_AJ0 run-length_NN1 for_PRP MIPS_NN0 ._SENT For_PRP comparison_NN1 ,_PUN we_PNP include_VVB expressions_NN2 for_PRP the_AT0 swim_NN1 pressure_NN1 put_VVD forward_AV0 in_PRP previous_AJ0 works_NN0 ._SENT Importantly_AV0 and_CJC in_PRP contrast_NN1 to_PRP Refs_NN2 ._SENT ,_PUN the_AT0 swim_NN1 pressure_NN1 derived_VVN using_VVG φ∗_NN1 does_VDZ not_XX0 asymptote_VVI to_PRP a_AT0 characteristic_AJ0 functional_AJ0 form_NN1 at_PRP large_AJ0 `_NP0 0._CRD In_PRP fact_NN1 ,_PUN the_AT0 dependence_NN1 on_PRP `_NP0 0_CRD is_VBZ quite_AV0 striking_AJ0 but_CJC not_XX0 unexpected_AJ0 given_VVN our_DPS newly_AV0 identified_VVN concentration_NN1 regimes_NN2 ._SENT 
