diff --git a/dynamics.py b/dynamics.py
index c3740d2..68f5a36 100644
--- a/dynamics.py
+++ b/dynamics.py
@@ -25,6 +25,13 @@ sys.path.insert(0, os.path.dirname(os.path.abspath(__file__)))
 import compute
 
 
+def _fmt_alpha(v):
+    """将 alpha 值格式化为逗号分隔字符串，兼容 numpy 数组/list/标量。"""
+    if isinstance(v, (list, tuple, np.ndarray)):
+        return ",".join(str(float(x)) for x in v)
+    return str(float(v))
+
+
 def read_optional_index(data, key, default_value):
     """Read an optional integer index from structured txt metadata."""
     if key not in data:
@@ -246,7 +253,7 @@ def run_case(config_path, runtime_base, input_dir="input", output_dir="output",
                   "gravity_strength": str(data.get("gravity_strength", 1.0)),
                   "driving_force": str(data.get("driving_force", 0)),
                   "use_marker": str(config.get("use_marker", 0)),
-                  "alpha": str(data.get("alpha", 0.2))}
+                  "alpha": _fmt_alpha(data.get("alpha", 0.2))}
             
             n_frames = len(indices)
             compute.save_display_txt(